Original ledger status
HYP.REWRITE.02
THE FULL TRANSFER SPECTRUM FROM ONE FORCED LOCAL RULE (2026-07-15, generator-hunt round; sharpens HYP.REWRITE.01/v327): the verified seam transfer spectrum {1, (2/3)^6, (1/3)^6} (v54/v221) now has a single local recursive generator -- the LAZY Z2-PAIR WALK. [E] UNIQUENESS: for the 3-channel rule M(s,h) = [[1,0,0],[0,s,h],[0,h,s]] (one absorbing family channel + symmetric Z2 pair) the survival spectrum is {s+h, s-h}; demanding the PHYSICAL pair {2/3, 1/3} forces UNIQUELY (stay, hop, leak) = (1/2, 1/6, 1/3) = (1/|Z2|, 1/(|Z2| N_fam), 1/N_fam) -- all three rates exact atom expressions (recorded; honest direction: spectrum -> rates). [E] SIX-HAND GENERATION: over the order-6 = 2 N_fam = |R+(A3)| dynamic hand (v327's hand) the rule generates EXACTLY eig(B^6) = {64/729, 1/729} = {(2/3)^6, (1/3)^6}; the two decay gaps are 6 ln(3/2) (the recovery gap, v76) and 6 ln 3 (the subdominant gap -- previously WITHOUT any local-rule origin). [E] SUPERSESSION: v327's uniform rule (s = h = 1/3, spectrum {2/3, 0}) has a persistent zero mode under every power -- no iterate can produce the (1/3)^6 mode; its lambda_2 content (2/3 = |Z2|/N_fam from the arity) is unchanged; superseded as generator of the FULL spectrum. [E] sub-stochastic, symmetric (detailed balance on the pair), absorbing democratic family attractor unchanged (v317). [O] HONEST RESIDUAL: the arity {|Z2|, N_fam} = {2, 3} stays the anchor input (as in v327); ADDITIONALLY the lazy split (1/2, 1/6) vs the uniform split (1/3, 1/3) is SELECTED by the physical spectrum, not yet derived from a first principle -- named candidate: 'stay = 1/|Z2|' as the sheet-symmetric choice of the Z2 involution (recorded, not claimed). Discovery provenance: experiments/tfpt-discovery/generator_hunt_full_transfer_rule.py (with the anchor-bilinear coverage inventory and the lepton-exponent (5,3,2) = K-row = atom-triple observation, both typed, no unforced claims). No gate closes, no marker moves. UPDATE 2026-07-15 (v487/HYP.REWRITE.03): the split selection [O] CLOSES -- the lazy rule's one-step spectrum is the complete v124 clock ladder below the wall and ladder faithfulness + Z2 equivariance + rates >= 0 force (1/2, 1/6) uniquely (the uniform rule collapses the odd channel onto the wall); only the arity and the clock's semiclassical derivation remain open.
Recorded status: Construction/uniqueness [E]/[O] (full spectrum from the unique lazy Z2-pair walk; split selection stays open)
Dependencies, status detail and source
Canonical status: Full-spectrum generator [E]: unique lazy Z2-pair walk (1/2, 1/6, 1/3); (1/3)^6 mode generator-full (gap 6 ln 3); v327 rule = degenerate case; arity [O]; split selection CLOSED by v487 (clock-ladder faithfulness)
Dependencies: HYP.REWRITE.01; HYP.FIBER.01; HYP.INJECT.01; DYN.SEMIGROUP.01; v54; v76; v221; v312; v317; v324; v327; v395
Scripts: v486_transfer_full_rule.py
Location: origin_theory
Original CSV, line 547Original ledger status
HYP.REWRITE.03
THE LAZY SPLIT FORCED BY THE RESUMMED CLOCK (2026-07-15, closing the v486 [O] 'split selection'; sharpens HYP.REWRITE.02): the split (stay, hop) = (1/2, 1/6) is not a choice -- it is clock-ladder faithfulness. [E] RUNG READING: the lazy rule's one-step spectrum is EXACTLY the complete v124 resummed-clock ladder below the wall, eig(M) = {1 - n/3 : n = 0, 1, 2} = {1, 2/3, 1/3}; v327's uniform rule instead puts its second pair mode ON the wall (0 = 1 - 3/3) -- the v486 degeneracy re-read as 'odd channel collapsed to the wall'. [E] DECK PARITY = RUNG INDEX: [B, sigma] = 0; the sigma-even pair mode carries rung n = 1 (survival 2/3), the sigma-odd mode rung n = 2 (survival 1/3) -- each clock step flips the deck parity; the structural home of the v486 parity assignment (omega_1 even-channel, omega_2 odd-channel) tested by the frb-parity-comb search. [E] FORCING: demanding the faithful ladder (no rung missing, no physical mode on the wall -- definitional for a full-spectrum generator) + Z2 equivariance (symmetric pair block) + non-negative rates forces UNIQUELY both the split (1/2, 1/6) AND the rung assignment even -> 1, odd -> 2 (the swapped assignment needs hop = -1/6 < 0, inadmissible); the v486 named candidate 'stay = 1/|Z2| as the sheet-symmetric choice' is superseded by this proof. [E] BEND COROLLARY: omega_1/omega_2 = rate(2)/rate(1) = log_{3/2} 3 = 2.70951... -- the two comb tones of the empirical search program are one v107/v124 bend apart. [O] HONEST RESIDUAL: the channel count 3 = N_fam is forced by the clock wall (v124 pt 4); the ladder itself stays an [I] identity on frozen spectra whose semiclassical derivation is open (R1/[P]); the arity {|Z2|, N_fam} = {2, 3} stays the anchor input. NET: no independent choice remains at the walk level. Contract provenance: experiments/theory-contracts/lazy01_clock_rung_forcing.py. No gate closes, no marker moves.
Recorded status: Construction/proof [E]/[O] (split forced by clock-ladder faithfulness; arity + clock derivation stay the open inputs)
Dependencies, status detail and source
Canonical status: Split forcing [E]: eig(M) = complete clock ladder {1, 2/3, 1/3}; uniform rule = odd mode on the wall; deck parity = rung index; assignment uniqueness via h >= 0; omega_1/omega_2 = bend; [O] arity + semiclassical clock derivation
Dependencies: HYP.REWRITE.02; HYP.REWRITE.01; v124; v107; v327; v486; v54; v221
Scripts: v487_transfer_clock_rungs.py
Location: origin_theory
Original CSV, line 548Original ledger status
FLAV.NUSCALE.04
THE 3/5 CLEBSCH DOOR ENUMERATED AND CLOSED (2026-07-15; executes the mechanism search behind FLAV.NUSCALE.03's named escape): v482 recorded that only a third-generation Majorana structure factor r ~ 1.6696 (0.18% from g_car/N_fam = 5/3) could revive the integer rung M_R = c3^3 Mbar, and declined it for lack of mechanism. The mechanism search is now EXECUTED at the group-theory level with a clean NEGATIVE. [E] NO RENORMALISABLE CHANNEL: sym^2(16) = 136 = 10 + 126; the 126bar is NOT in the E8 hull {1, 10, 16, 45} (v247) -- M_R only via the d=5 operator (16_F 16bar_H)(16_F 16bar_H)/Lambda with the E8-allowed 16_H. [E] CHANNEL LIST: 16 x 16bar = 1 + 45 + 210; the hull admits singlet + 45 insertion channels (210 absent). [E] CHARGE TABLE: Y = T_3R + (B-L)/2 verified on all 16 states; Tr_16 Y^2 = 10/3 (v485); (B-L, T_3R, Y)(nu^c) = (+1, -1/2, 0). [E] THE {2,3}-SMOOTH THEOREM (the door closes): every available nonzero nu^c channel weight -- singlet 1, T_3R^2 = 1/4, (B-L)^2 = 1, T_3R(B-L) = -1/2, SU(4)-15 weight 3/8 -- is a {2,3}-smooth rational, and the multiplicative closure of {2,3}-smooth rationals never contains 5/3 (a factor 5 cannot appear); the UNIQUE natural 5 of the embedding is k_Y = Tr_16 Y^2 / Tr_16 T_3L^2 = 5/3 (Ginsparg, v470) -- a FULL-MULTIPLET trace, and its carrier direction has Y(nu^c) = 0 EXACTLY, so the only 5-carrying group number is structurally decoupled from the Majorana operator. [E] GENERATION-BLINDNESS: the Clebsch multiplies the whole family matrix c_ij (identical 16's); a per-generation ratio is a flavor texture, never a Clebsch -- the required diag(1, 1, 3/5) cannot arise from group theory by construction. [O] COMPILER-FLAVOR INVENTORY (typed, declined): exact 3/5's in the frozen address tables sit only in WRONG slots (K lepton row (5,3,2): gen2/gen1; R (2,5,3) and L (8,5,3) lepton rows: gen3/gen2); the escape needs gen3 vs a DEGENERATE gen1 = gen2 base, which no frozen table has; no mechanism links M_R to the address tables (v354/v355). NET: the rung + 5/3 rescue is DEAD as a mechanism candidate; the v481 one-parameter window candidate (M_R free inside the PS window; Sigma m_nu = 0.059 eV floor as kill test) stands as the honest endpoint. Contract provenance: experiments/theory-contracts/clebsch01_majorana_35_door.py. No gate closes; the frozen record is untouched.
Recorded status: Enumeration/decision [E]/[O] (the 3/5 Clebsch door closed; negative result; window candidate stands)
Dependencies, status detail and source
Canonical status: Door closed [E]: no renormalisable channel (126 E8-forbidden); nu^c channel weights {2,3}-smooth -> 5/3 unreachable; k_Y = 5/3 decoupled (Y(nu^c) = 0); Clebsches generation-blind; [O] wrong-slot compiler 3/5's typed and declined
Dependencies: FLAV.NUSCALE.03; FLAV.NUSCALE.02; PS.E8BRANCH.01; v247; v249; v470; v481; v482; v485
Scripts: v488_majorana_clebsch_door.py
Location: tfpt_2_standard_model
Original CSV, line 549Original ledger status
FLAV.NUSCALE.05
THE SCALARON-TRACE RUNG CANDIDATE (registered 2026-08-28, external master-object review wave 2): M_R = N_fam * M_scal = 3 c3^{7/2} Mbar = 9.180e13 GeV -- a STRUCTURED point inside the v481 PS window, -1.8% from the v481-required 9.346e13 GeV under the same frozen 1-loop inversion and the named premise y_nu = y_t [P] (the review text said about 1%; the honest value is 1.8); m_3 at the candidate = 0.0512 eV (+1.9%, inside the >50% v482 RG envelope), ~20x sharper than the declined integer rung c3^3 Mbar. EXACT ladder position: 3 c3^{7/2} = (3 sqrt c3) c3^3, 3 sqrt(c3) = 3/sqrt(8 pi) = 0.598413 -- 0.26% from the 3/5 of the v482/v488 story; ADJACENCY TYPED: v488 killed 3/5 as an SO(10)/PS CLEBSCH, while the scalaron-trace reading is a DIFFERENT mechanism class -- HONEST RETYPE (2026-08-28, review wave 3): the trace argument Tr_A3 I = 3 is loose (Spec(I) = {1,1,1}, trace 3 does not make 3 an eigenvalue); the operator reading is the Q_+ eigenvalue route Spec{1,2,3} (v10/v50/v69), with the mixed insertion texture typed DATA_CONSTRAINS_TEXTURE in the nu-scalaron v2 project -- mechanism stays [O], not derived. The same M_scal already carries f_a = M_scal/128 (v185) and the inflation scale (v253): no new scale atom. KILL TEST: Sigma m_nu (NO) = 0.059 eV; a robust cosmological bound below 0.059 eV or a >~5% shift of the required M_R kills the candidate. CANONICAL NOTE (2026-08-28, completeness wave, NU_TEXTURE_CENSUS_NULL, not a module): frozen 3x3 inventory search 0/1607 corpus candidates + 0/200 random control (best miss factor 12.6 on the reactor angle) -- the Q+ union needs structure beyond the frozen 3x3 inventory; mechanism [O] sharpened. Provenance: experiments/tfpt-discovery/nu_texture_census_probe.py. No v3 freeze; M2 = 1.73e10 GeV remains the unused prediction.
Recorded status: Candidate [C]/[N] + mechanism [O] (structured point inside the v481 window; scalaron-trace hypothesis typed, not derived; no seesaw closure)
Dependencies, status detail and source
Canonical status: [N] 3 M_scal within 1.8% of the required M_R and m_3 within 1.9% at the frozen v481 inputs; [E] exact ladder identities; [C] the scalaron-trace reading; [O] the mechanism; [X] the Sigma m_nu = 0.059 eV kill floor. Does NOT close FLAV.NUSCALE; v481 window candidate and v488 negative untouched. WAVE-3 RETYPE (2026-08-28): operator reading is the Q_+ eigenvalue route Spec{1,2,3} (v10/v50/v69), not Tr I = 3; texture DATA_CONSTRAINS_TEXTURE (nu-scalaron v2); mechanism stays [O]. COMPLETENESS-WAVE NOTE (2026-08-28): NU_TEXTURE_CENSUS_NULL 0/1607 + 0/200 -- frozen 3x3 inventory provably insufficient; mechanism [O] sharpened; no new module (search null). WAVE-5 (2026-08-29, v1003): Schur-texture NULL (8/8 D4-even (K,B) fail; lambda_1=lambda_3 forced) and scale NULL (12/12; 1-|c/a|=1.885e-4 not supplied) recorded as binding constraints on FLAV.NU.TEXTURE.MECHANISM.01 [O]. Candidate unmoved; no seesaw closure.
Dependencies: FLAV.NUSCALE.02; FLAV.NUSCALE.03; FLAV.NUSCALE.04; GRAV.FR.01; AX.P1.01
Scripts: v986_nu_scalaron_rung.py, v1003_nu_structural_nulls.py
Location: tfpt_2_standard_model
Original CSV, line 550Original ledger status
SEAM.CCC.01
THE CHIRAL CENTRAL CHARGE c_- = 8 MEASURED FROM THE BULK GROUND STATE (2026-07-20; ground-state witness round): the Kim-Shehab-Kim modular commutator J(A,B,C) = i Tr(rho_ABC [K_AB, K_BC]) = (pi/3) c_- (PRL 128 (2022); arXiv:2110.06932/2110.10400), evaluated on the EXACT BdG ground state of the v367 p+ip seam lattice model via the Peschel Majorana covariance (W_X = -2 artanh Gamma_X, J = (1/4) Tr([W_AB, W_BC] Gamma_ABC)), a disk about the torus centre tripartitioned into three 120-degree wedges. This is the FIRST ground-state witness for premise (i) of SEAM.EQUIV.01 (c_- = 8) that is INDEPENDENT of both the Chern-number route (v367: FHS curvature of the Bloch bundle) and the counting route (v378: KLM/det-Cartan arithmetic) -- J is read off entanglement data alone; no band topology or representation theory enters the measurement. [E] GROUND-STATE PURITY: eig(i Gamma) = +-1 to 6e-15 (exact pure state, not an approximation). [E] PER-LAYER c_- = 1/2: c_- = 3J/pi = 0.499998 at L=32, R=12 (convergence 0.499793 at L=16 -> 0.500000 at L=48, R=18, dev -1.8e-7; suite runs L=32 with tol 1e-4). [E] CONTROLS: trivial M=3 (Chern 0) gives c_- = 0.000000; orientation flip M=-1 gives -0.499986 (sign flip) -- J tracks the phase, not the algorithm. [E] LAYER ADDITIVITY EXACT: J(2 decoupled layers) - 2 J(1 layer) = 1.4e-14 (block-diagonal Gamma), so the carrier value is 16x the per-layer value. [E] THE 16-LAYER CARRIER: c_- = 8 = g_car + N_fam -- direct 16-layer computation 7.9967 (L=16, tol 1e-2; the L=24 direct run gives 7.999781) and additivity route 8.0000 (tol 1e-3). [E] CLIP INVARIANCE: the artanh regularisation eps = 1e-8 vs 1e-12 moves c_- by 5e-8. [O] HONEST FENCE: this pins c_- = 8 on the LATTICE ground state (third independent witness after v367/v378); it does NOT close SEAM.EQUIV.01 -- the continuum scaling limit (MMST, v336) stays [O] and the mu4 condensation SO(16)_1 -> (E8)_1 stays the cited KLM step (v378/v154/v351). No gate closes, no marker moves.
Recorded status: Lattice [E] (modular-commutator ground-state measurement + controls + exact additivity) + [O] (continuum limit; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: Ground-state witness [E]/[O]: c_- = 0.499998 per layer (0.500000 at L=48), 16-layer carrier c_- = 8 = g_car + N_fam (direct 7.9967 / additivity 8.0000), trivial 0 and orientation-flip controls, exact layer additivity -- premise (i) of SEAM.EQUIV.01 pinned from entanglement alone; SEAM.EQUIV.01 stays [O]
Dependencies: SEAM.EQUIV.01; SEAM.S3.LATTICE.01; SEAM.S3.MODULAR.01; SEAM.MMST.INCLASS.01; v336; v351; v367; v368; v376; v377; v378
Scripts: v489_seam_modular_commutator.py
Location: tfpt_1;tfpt_research_contracts
Original CSV, line 551Original ledger status
SEAM.MU.01
THE SPIN-STRUCTURE PARITY CENSUS: mu_pre = 4 WITNESSED ON THE LATTICE (2026-07-20; ground-state witness round, companion of SEAM.CCC.01/v489): the T^2 ground-state fermion parity of the v367 p+ip seam lattice model, resolved per spin structure (boundary-condition sector AA, AP, PA, PP) and determined TWICE -- (a) the Read-Green TRIM product parity(s) = prod_{k* in TRIM(s)} sign(dz(k*)) and (b) the real-space Majorana Pfaffian sign via Schur decomposition (parities relative to the trivial M=3 phase so the Pf convention drops out). The number of even sectors = the gauged (fermion-parity-gauged) torus GSD = the total quantum dimension mu of the gauged theory. [E] REAL-SPACE = BLOCH SANITY: the twisted real-space BdG spectrum equals the Bloch bands per sector to 9e-15. [E] PER-LAYER CENSUS: (AA, AP, PA, PP) = (+,+,+,-), TRIM product == Pfaffian in every sector -- 3 even sectors, the nu = 1 Ising signature (gauged #anyons = 3). [E] 16 LAYERS => mu_pre = 4: parity_16(s) = parity_1(s)^16 = + in ALL FOUR sectors => gauged torus GSD = 4 = mu(SO(16)_1) = |det Cartan(D8)| -- the lattice ground-state witness that the carrier sits in the SO(16)_1 class BEFORE the mu4 condensation (v378 step 1). [E] 16-FOLD-WAY BRIDGE: Kitaev's theta_v = exp(2 pi i nu/16) with the MEASURED nu = 2 c_- = 16 (v489) gives theta_v = 1 EXACTLY (sympy) -- the gauged vortex is a boson, condensable; rivals nu = 1, 2, 8 give theta_v != 1 (not holomorphically condensable). [E] NON-DISCRIMINATOR FENCE (honest): the fermionic Kitaev-Preskill TEE is identically zero -- gamma_f = 0.000001 (M=1, topological) AND 0.000000 (M=3, trivial) -- an invertible fermionic phase has no TEE, so gamma_f does NOT discriminate (E8)_1 vs SO(16)_1 (both exist only after gauging); computed and recorded so it is never mistaken for a discriminator. [C] THE KLM CONDENSATION STEP stays cited: mu_post = mu_pre/|L|^2 = 4/2^2 = 1 (holomorphic, (E8)_1; v378/KLM CMP 219 (2001)); the condensed model is interacting, NOT free-fermion computable -- the numerics pin only the premise package {c_- = 8 (v489), mu_pre = 4, theta_v = 1}. [O] the continuum scaling limit (MMST, v336) stays [O]; SEAM.EQUIV.01 is NOT closed. No gate closes, no marker moves.
Recorded status: Lattice [E] (parity census double-determined + exact 16-fold-way arithmetic + TEE fence) + [C] (KLM condensation step cited) + [O] (continuum; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: Lattice witness [E]/[C]/[O]: parity census (+,+,+,-) per layer, all-even for 16 layers => gauged torus GSD = 4 = mu(SO(16)_1) = mu_pre; theta_v = 1 exactly at nu = 2 c_- = 16 (rivals excluded); TEE fence gamma_f = 0 for both phases (no discriminator); KLM 4 -> 1 stays [C]; SEAM.EQUIV.01 stays [O]
Dependencies: SEAM.EQUIV.01; SEAM.CCC.01; SEAM.S3.LATTICE.01; SEAM.S3.MODULAR.01; v336; v351; v367; v378; v489
Scripts: v490_seam_parity_census.py
Location: tfpt_research_contracts
Original CSV, line 552Original ledger status
P2.PARTITION.01
g_car = 5 AS A COROLLARY OF THE FOUR MARKS (2026-07-20; the partition sharpening of ARCH.CORE.01/v53): a multiset of exactly 3 POSITIVE INTEGERS summing to 4 is UNIQUE -- {1,1,2}, the unique partition of 4 into exactly 3 positive parts -- so the elementary symmetric functions e = (4,5,2) and hence g_car = e2 = 5, |Z2| = e3 = 2, N_fam = e2 - e3 = 3 are COROLLARIES, not inputs; the carrier formula closes as a fixed point ((2^(g_car-1)-1)/g_car = 3 = rank H^1 = #marks - 1). [E] NEGATIVE CONTROLS, each assumption load-bearing (sympy exact): drop positivity -> 4 solutions with e2 ambiguous {0,3,4,5}; sum 3 -> unique but e2 = 3; sum 5 -> not unique; 2 parts -> e2 in {3,4}; 4 parts -> e2 = 6; Mehta-Seshadri-typed RATIONAL weights (scaled by 4) -> e2 = 16/3 resp. 44/9 != 5 -- integrality comes from the SPLITTING-TYPE typing (Birkhoff-Grothendieck: bundles on P^1 split with integer degrees), NOT from MS weights (which are rational). [E] THE v53 SHARPENING: v53 pins (1,1,2) using ALL THREE targets (4,5,2); the sharper lemma needs only e1 = 4 (+ positivity + 3 parts) -- both searches return the same unique multiset. [E] THE SUM RULE: |a|_1 = 4 x (N-1)/2 = 4 iff N = 3 (equidistributed clock spectrum); the v115 mu4-average route |a|_1 = |mu4| tr(A0) = 4 agrees. INDEPENDENCE CHAIN (why not circular in g_car): #marks = 4 from v216 Gauss-Bonnet (P1-side premises only, checks 1/3/5 without g_car); the 3 = rank H^1(P^1 minus 4 marks) = #marks - 1 is pure topology (v137 explicit basis / v228 index gate); |a|_1 = 4 is g_car-free via par-deg-0/(U) (v115) or the mu4-average lemma. [C] HONEST RESIDUAL: P2 is NOT made axiom-free -- the residual is the weight-TYPING postulate (integrality = Birkhoff-Grothendieck splitting exponents; positivity a_i >= 1 = h^0(E) = 0, correlate: the trivial mu4 character absent from H^1, v137 check 1; sum rule = parabolic-degree-0 'given (U)', v115 [I]/[N]); AX.P2.01 stays a DECLARED AXIOM (over-determined, now by one more convergent route). DIRECTION-REVERSAL NOTE: tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> partition -> g_car -- an over-determination check, not a circle, and it must never be double-counted in witness tallies. No gate closes, no marker moves.
Recorded status: Formal [E] (partition lemma + corollaries + negative controls + v53 sharpening, sympy exact, Wolfram-mirrored) + [C] (weight-typing postulate: splitting-type integrality, h^0 = 0 positivity, (U) sum rule; AX.P2.01 stays Axiom)
Dependencies, status detail and source
Canonical status: Partition corollary [E]/[C]: 4 = 1+1+2 unique in exactly 3 positive parts => g_car = 5, |Z2| = 2, N_fam = 3 as corollaries; all negative controls pass; sharpens v53 (only e1 = 4 needed); residual = the weight-typing postulate -- AX.P2.01 stays a declared axiom; the typing residual is itself sharpened by v499 (P2.TYPING.01, Deligne + BG + stability), v491 stays active
Dependencies: AX.P2.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; v23; v53; v115; v137; v216; v228
Scripts: v491_p2_partition_corollary.py
Location: introduction;tfpt_1;tfpt_2;origin_theory
Original CSV, line 553Original ledger status
P2.TYPING.01
THE P2 WEIGHT-TYPING POSTULATE HARDENED (2026-07-21; types the v491/P2.PARTITION.01 check-9 residual): the anchor weights a = (1,1,2) are the Birkhoff-Grothendieck SPLITTING EXPONENTS of the DELIGNE CANONICAL EXTENSION E of the flavor connection -- given the [E] anchors (4 marks, v216 Gauss-Bonnet; rank 3 = #marks - 1, v137/v228; cusp-class local monodromy char poly lam^3 - 1 + irreducibility |<U,M0>| = 24 = |S4| with sum |chi|^2 = 24 exact, v117; unitarity (U)), the three typing properties become THEOREMS of standard algebraic geometry: INTEGRALITY = Birkhoff-Grothendieck (every holomorphic bundle on P^1 splits with integer degrees, automatic once E is typed as a bundle); deg E = -4 = -#marks via the Deligne residue-trace formula (canonical residue exponents {0,1/3,2/3}, the UNIQUE lift of spec M0 to [0,1), trace 1 per mark; par-deg 0 = the v39 (U) consistency; infinity APPARENT, monodromy product = 1 exactly); POSITIVITY a_i >= 1 = h^0(E) = 0 via Mehta-Seshadri stability (irreducible + unitary => stable; every section-saturation has par-deg >= 0, all 324 cases; BG dictionary h^0 = 0 <=> a_i >= 1 exhaustive) -- so {a} = {1,1,2} is UNIQUE (v491 partition lemma) and g_car = e2 = 5 follows. [E] SELECTOR TYPED: the Schur-Horn permutohedron of 4 spec(A0) has integer points exactly {(2,2,0),(2,1,1)} (the two options of the tfpt_2 'Global computation' theorem); h^0 = 0 kills {0,2,2} (O(0) summand => h^0 = 1) -- the paper's 'balanced/minimal-variance' heuristic is now TYPED as stability/positivity. [E] THE h^1 DISCREPANCY, REAL + RESOLVED: h^1(O(-2)+O(-1)^2) = 1 != 3 (chi = deg + rk = -1) -- the naive typing 'generation space = H^1 of the rank-3 E' is FALSE (forcing it demands deg -6 and loses uniqueness: {1,1,4},{1,2,3},{2,2,2}, e2 in {9,11,12} never 5 -- negative control); the correct TWO-OBJECT typing: the LINE bundle L = O(-D), D = mu4, carries the generations (h^1(L) = 3 = N_fam = H^0(Omega^1(log D)) = the v137 basis; h^0(O(4)) = 5 = g_car, v228), while E carries them as its FIBER (rank 3) with the anchor as splitting type; three DIFFERENT H^1's coexist (3 generations L-side, 1 coherent h^1(E), 6 = |R+(A3)| local-system deformations). [E] BISWAS EXPLICIT (independent convergence witness): the weight denominator is 3 = N_fam (NOT 2 = |Z2|), so the Biswas cover is the Z3 cusp cover y^3 = (z-1)(z-i)(z+1)^2(z+i)^2 (genus 2 by RH, unramified over infinity; all-equal branch data impossible, 4c != 0 mod 3), NOT the pillowcase double w^2 = z^4-1 (genus 1, j = 1728, deck side); p_* O_Y = O + O(-2)^2 (chi cross-check -1 = -1); the equivariant regular-rep model W = (chi^0+chi^1+chi^2) x O_Y has parabolic partner with underlying bundle O + O(-2)^2: rank 3, deg -4, par-deg 0 all OK -- integrality AND the sum rule COME OUT of the correspondence -- but splitting {0,2,2} with h^0 = 1: exactly the UNSTABLE Schur-Horn companion; (1,1,2) is the Biswas image of the STABLE object, stability is the selector. [E] NEGATIVE CONTROLS: drop positivity -> 4 (31 with negatives) solutions; deg -3 -> e2 = 3, deg -5 -> not unique; n-mark scan: {1,..,1,2} unique for EVERY n >= 3 but e2 = (n-2)(n+1)/2 = 5 ONLY at n = 4 (the VALUE, not the uniqueness, is the n = 4 content); MS-rational exponents e2 = 16/3 resp. 44/9. [C]/[N] HONEST RESIDUALS (P2 stays an axiom, NO marker moves): R1 the module identification 'generation space = this H^1 with this connection' [C] (QGEO.SYM territory, QGEO.SYM.01 stays a corollary of the [O] keystone); CONVERGENCE THEOREM (2026-07-21, v503/QGEO.EMERGE.LIGHT.01): R1 is IDENTICAL with the QGEO modulus rest -- the order-4 clock on H^1 is the Coxeter element of W(A3) = the cusp-class carrier datum, and the QGEO emergence-light test shows marks + square are derived while the clock is the one input: QGEO-R1 == P2-R1, ONE common residual, not two; FERMIONIC HALVING (2026-07-21, v506/SEAM.CLOCK.RIGIDITY.01): the common R1 rest is mapped onto the SAME single alignment bit -- 'the collar deck is the central involution of the mark-D4' -- because the ORDER of the clock is now fermionically forced (U^2 = (-1)^F exact on the NS seam, nonsplit Z4; both Moebius square roots automatically order 4), leaving only the spatial alignment as the carrier input; ORIGIN CROSS-REFERENCE (2026-07-21, v507/SEAM.BIT.ORIGIN.01 -- marker UNCHANGED): the common R1 bit is now characterised as a tetrad with a physical face (non-split NS Fock extension class) and its deck CLASS is derived (half-period translation of the seam double, v507); only the position stays the carrier input; FREEDOM HALVING (2026-07-21, v510/SEAM.BIT.FREEDOM.01 -- marker UNCHANGED): the common R1 bit is HALVED -- its position half is topology (the collar deck is a covering deck, hence free on the seam circle; the edge class is excluded, v510), so R1 reduces to the square-modulus datum tau = i alone; the modulus stays the one [C] carrier input; R2 unitarity (U) [C] (v33/v40 premise); R3 the analytic-monodromy identification [N] (v117 check 4, ODE matching 1e-7). NON-CIRCULAR: neither v23's splitting-type input nor v115's diag(A0) = a/4 input is used; the chain is rank (topology) -> deg (residue trace) -> integrality (BG) -> positivity (stability) -> partition (v491); A0* appears only as the explicit consistency witness. The weight postulate shrinks from 'positive integers summing to 4 (why?)' to 'Deligne + BG + stability, given the cohomological typing'. DIRECTION-REVERSAL NOTE inherited from v491: tfpt_constants computes N_fam FROM g_car; here the arrow runs marks -> bundle -> partition -> g_car -- an over-determination check, not a circle, never double-counted. AX.P2.01 stays a DECLARED AXIOM. SYNTHESIS NOTE (2026-07-22, v512/SEAM.TAU.FLAG.01 -- marker UNCHANGED): the reduction state reads 'four marks + one discrete symmetry-lift bit (flag transitivity <=> tau = i) + pi' -- the R1 rest gains an equivalent discrete seam-intrinsic form (V4 -> D4 side-indistinguishability of the four equally-decorated marks), not a derivation.
Recorded status: Formal/Identity [E] (AG arithmetic: Deligne residue-trace degree, BG dictionary, stability skeleton, Schur-Horn lattice points, Biswas Z3-cover -- sympy exact, Wolfram-mirrored) + [C] (R1 module identification, QGEO.SYM territory; R2 unitarity (U)) + [N] reference (R3 analytic monodromy, v117 check 4); AX.P2.01 stays Axiom
Dependencies, status detail and source
Canonical status: Weight typing hardened [E]/[C]: rank 3 (topology) + deg E = -4 (Deligne) + integrality (BG) + positivity (MS stability, h^0 = 0) force E = O(-1)^2 + O(-2) => a = {1,1,2}, g_car = 5; balanced heuristic typed as stability; h^1 discrepancy resolved (two-object typing); Biswas Z3-cover explicit; residuals R1/R2 [C], R3 [N]; R1 mapped by v506 onto the single alignment bit (deck central in the mark-D4; the clock ORDER is fermionically forced); v510 halves R1: the position half is topology (edge class excluded), the rest = the square-modulus datum tau = i; v512 restates the rest as one discrete symmetry-lift bit (flag transitivity <=> tau = i) -- AX.P2.01 stays a declared axiom, QGEO.SYM stays [O]
Dependencies: AX.P2.01; P2.PARTITION.01; ARCH.CORE.01; ANCHOR.GEN.01; MARKS.GB.01; QGEO.SYM.01; v33; v40; v115; v117; v137; v216; v228; v491; QGEO.EMERGE.LIGHT.01; v503; SEAM.CLOCK.RIGIDITY.01; v506; SEAM.BIT.ORIGIN.01; v507; SEAM.BIT.FREEDOM.01; v510; SEAM.TAU.FLAG.01; v512
Scripts: v499_p2_weights_deligne_bg.py
Location: introduction;tfpt_1;tfpt_2;origin_theory
Original CSV, line 554Original ledger status
CELEST.WP1.01
WP1 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 1, verdict B): the E8 mu4-glue grading (v128) IS the flat Z4 monodromy of the glue-equivariant celestial chiral algebra on the A3 ALE space C^2/Z4 -- the Bittleston-Homans-Sharma CCA construction (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case) refined to Z4 with a Kronheimer-Nakajima flat connection at infinity. Sympy exact throughout. [E] INNER GRADING / FLAT MONODROMY: h = (2,2,2,2,2;0^4) gives <alpha,h> = glue class mod 4 for ALL 240 roots => glue = Ad(exp(2 pi i h/4)), an order-4 torus element = a flat Z4 monodromy; the A3-side detector h' = (0^5;1,1,1,-3) reads the SAME class -- the glue charge is the diagonal (1,1) of the discriminant group Z4 x Z4 (the v92/v125 Lagrangian glue); Z2 halfway: 120 = SO(16), 128 = spinor (v125 sub-Q {0,2}). [E] SPATIAL SIDE: C^2/Z4 under (z1,z2) -> (i z1, i^-1 z2) IS the A3 singularity XY = Z^4 (invariant ring exact; Hilbert series (1+t^4)/((1-t^2)(1-t^4)); Molien table verified; conjugation symmetry N[d][1] = N[d][3]). [E] EQUIVARIANT SECTOR CLOSES: the glue-equivariant SDYM(E8) modes (j + m = 0 mod 4) form a Lie subalgebra with D[d] = 60(d+1) even / 64(d+1) odd -- uniform per parity ONLY because dim g_j = (60,64,60,64) (the v128 dims are the consistency condition); zero modes (d=0) = g0 = carrier D5+A3+Cartan = 60; asymptotic density 62/248 = 1/4 = 1/|Z4|; EH/Z2 halfway towers 120/128. [E] CHARACTER: Theta_E8 = sum of the 4 glue-coset thetas (shell 1 = (52,64,60,64) from the LATTICE, independent of the root construction); the 4 sector characters Theta_Cj/eta^8 sum EXACTLY to chi_(E8)_1 = 1+248q+4124q^2+34752q^3 (v377 series); sector weights h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8) = the v92/v125 discriminant form (5x^2+3y^2)/8 mod 1; glue diagonal h = (0,1,1,1) INTEGER = v125 isotropy = the extension to (E8)_1 is local. [E] CRITICAL CORRECTION (why verdict B): the A3 deck diag(i,i^-1) induces z -> -z on the celestial sphere (projective order 2, the sheet flip) -- NOT the order-4 clock z -> iz; any SU(2) lift of the clock has order 8 (spin group Z8, would quotient to A7); exact bridge (spin clock)^2 = deck, |Z8| = 8 = 2|mu4| (the c3 = 1/(8 pi) winding integer); the order-4 clock IS realised as the U(2) phase diag(i,1) (order 4, fixed {0,inf}, free mu4 orbit -- all v180 properties; normalises the deck, det = i rotates the holomorphic symplectic form by the mu4 generator -- Kaehler, not triholomorphic). [E] GRAVITY SIDE: clock-invariance selects from the versal A3 family XY = Z^4 + a2 Z^2 + a1 Z + a0 EXACTLY the 1-parameter family XY = Z^4 + a0; its 4 branch points are ONE mu4 orbit with cross-ratio 2 -- the seam marks (v179 clock orbit; v168/v214 pillowcase j = 1728). [E] NEGATIVE CONTROLS: false spatial action diag(i,i) killed 3 ways (det = -1 not in SL(2,C); Poisson closure fails 14/14; grades unsymmetric); false glue (60/64 swap) breaks additivity on 3112/6720 root pairs; rigidity: exactly {id, j -> -j} = Aut(Z4) of 24 relabelings are additive. [C] the clock-deck identification is a NAMED extra identification (clock^2 = deck on the sphere, sheet-Z2 spin bookkeeping) -- verdict B, not A. [O] structural arithmetic ONLY: no celestial duality is constructed; CELEST.SEAM.01 (WP2-WP5) stays a research contract; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact Z4-orbifold/character/spin arithmetic, sympy, negative controls) + [C] (the named clock^2 = deck identification, verdict B) + [O] (celestial programme; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP1 [E]/[C]/[O]: E8 mu4-glue = inner order-4 torus element = flat Z4 monodromy on the A3 ALE; equivariant SDYM(E8) sector closes (60/64 towers, zero modes = carrier, density 1/4); (E8)_1 character = sum of the 4 glue sectors, weights = discriminant form; clock = U(2) diag(i,1), clock^2 = deck (Z8, 8 = 2|mu4|); clock-invariant deformation XY = Z^4 + a0, branch points = one mu4 orbit, cross-ratio 2; negative controls + Aut(Z4) rigidity; verdict B; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; SEAM.EQUIV.01; E8.GLU.01; v92; v125; v128; v168; v179; v180; v214; v377
Scripts: v492_celestial_z4_orbifold.py
Location: tfpt_research_contracts
Original CSV, line 555Original ledger status
CELEST.SEAM.01
THE CELESTIAL-HOLOGRAPHIC RESEARCH CONTRACT (2026-07-20; the fourth research contract alongside CONTRACT.U.01, CONTRACT.G.01 and CONTRACT.F.01): HYPOTHESIS (as corrected by WP1) -- the seam is the glue-equivariant Z4-orbifold sector of the celestial chiral algebra on the A3 ALE space C^2/Z4; the mu4 clock is the Kaehler U(2) phase diag(i,1) whose square is the deck group (spin bridge Z8, 8 = 2|mu4| = the c3 = 1/(8 pi) seam winding integer); the associated twistorial bulk is the SELF-DUAL sector of TFPT gravity. TYPING / NON-CIRCULARITY: admissible inputs = the finite seam/glue data (v92/v125 glue, v128 graded hull, v168/v214 pillowcase, v180 clock) + the WP1/WP2 [E] arithmetic (v492/v493); NOT admissible as input: the continuum existence of the holomorphic (E8)_1 net on the seam (exactly the SEAM.EQUIV.01 target -- assuming it would be circular); scope = the self-dual sector only, never full Einstein gravity (the v358/v359 entanglement-equilibrium route and GRAV.ENTROPIC.ACTION.01 are separate statements about the full equation). WORK PACKAGES: WP1 DONE (v492, verdict B -- match under the one named identification clock^2 = deck); WP2 DONE (v493, verdict B -- both targets landed and SHARPENED: a0 = pure seam SCALE, the tau = i pillowcase shape frozen (I = 12 a0, J = 0 identically, j = 1728 for all a0); the 3 spheres carry the 3 nontrivial mu4 characters, the clock = Coxeter element of W(A3) = the Picard-Lefschetz monodromy of the family, surviving direction = the chi_1-Fourier diagonal; BHS Z2 -> Z4 transfer with corrections linear in a0, grade-conserving, a0 <-> the c^2 slot); WP3 DONE (v495, verdict B -- exact arithmetic alignment EXISTS: lambda_e8 = 6 (unit-trace) resp. 1/10 (adjoint-trace), (kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300, the Okubo coefficient 5/(2(dim+2)) DERIVED for all 8 algebras with a sharp sl4 negative control and the closed form lambda~^2 = h_vee + 6 across the Deligne series; BUT the look-elsewhere battery shows the alignment format passes 8/8 -- ZERO selectivity; only g_car = 5 | h_vee is e8-selective (1/8), lambda-integrality 2/8 shared with sl3, the isolating conjunction POST HOC: the c3-connection survives as convention-level compatibility + genuine lambda-arithmetic, NOT as e8-selective evidence -- alignment survives, selectivity does not); WP4 DONE (v496, verdict B(ii) -- the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading, the mismatch localised THREE ways (spin unbounded below; growth n^{2/3} vs n^{1/2}, Meinardus poles s = 2 vs 1; the level-2 null ideal deletes exactly 27000 = 30^3 = h_vee^3 with no jet analogue), BUT the boundary/period reading holds exactly at the current stratum (zero modes = 60 = C0 currents, jet slice cycles (60,64,60,64), one mu4 period sums to 248, glue weights (0,1,1,1) integer) -- the SAME MMST scaling-limit structure as SEAM.EQUIV.01 (v336/v449): the character is a boundary/limit SHADOW of the S-algebra, the constructive limit question goes to WP5); WP5a DONE (v497, 2026-07-21 -- 'the null ideal from the limit': the boundary limit made a PRECISE coefficientwise limit via the chi_w family with stabilisation threshold w = n+1 (w = 2 = the chiral jet grading); the null ideal DERIVED from root data, Sym^2(248) = 27000 + 3875 + 1 with 27000 = h_vee^3 by Freudenthal + Weyl + peeling; quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control: four blocks, 5304 != 14^3, h = 1/2 breaks one-block fusion; the celestial route now has the SAME two-step shape as MMST -- limit + maximal ideal -- with the ideal quantitatively fixed); WP5b DONE (v498, 2026-07-21 -- 'the singular vector as an operator', SUCCESS on the preregistered criteria: |s> = (E^theta_{-1})^2|0> constructed EXPLICITLY in a machine-built Chevalley/Frenkel-Kac basis (cocycle asymmetry on all 57600 pairs, SGN = -1 forced by Jacobi, kappa derived), J^a_1|s> = 0 for all 248 generators with case tally 190/57/1 (exactly one case sees k), level dial 2(1-k): the deletion object exists ONLY at k = 1 and not in the centerless loop algebra; mu4: j(theta) = 1 (h-adapted chamber), clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 with the direct g_0-orbit BFS matching Freudenthal through depth 4, quotient 4124 = sector sum; controls: the 248-current state and generic level-2 states NOT singular, the cube at k = 2 (generic Kac mechanics, honestly typed), D8 has the singular vector too but h = 1/2 breaks one-block fusion -- one-block closure stays E8/mu4-specific; HANDOVER to WP5c: in the twisted quarter-slot moding two C_1 modes never sum to 8 quarters (minimum 6 = q^{3/2}) -- the per-period dictionary (v496), not the per-slot identification, carries |s> to q^2); WP5c DONE (v500, 2026-07-21 -- 'the GNS limit state', SUCCESS on the preregistered criteria: the quasi-free family omega_w (affine k=1 vacuum on the currents via the machine-determined compact anti-involution theta(e_a) = -e_{-a}, unique on all 61504 basis pairs, + x^{wr}-contracted radial oscillators) is positive for every finite w and stabilises EXACTLY at the WP5a threshold (omega_w = omega_inf mod x^{N+1} for w >= N+1, sharp at w = N); GNS(omega_inf) levels (1, 248, 4124) via the FULL exact 9361-block level-2 Gram: rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight (Freudenthal on all 9361 weights), every block PSD, rank table per Weyl orbit (0,0,1,8,44), h.w. norms (3844,49,2,0); the clock descends to GNS with level-2 rank split (1036,1024,1040,1024) = the per-period lattice theta-split -- the two-routes identity at the STATE level; |s> IS the zero vector of GNS(omega_inf), resolving the WP5b S4.10 twisted-slot tension; CCR obstruction: NO w-uniform damping state exists -- the family formulation is necessary AND the family exists, KILL not triggered; controls: k=2 keeps everything, k=0 has no current layer, D8 gives 2076 = one block of four, wrong family erases the 248 layer 710955 != 248, no damping keeps 897266 != 248); WP5d-alpha DONE (v501, 2026-07-21 -- 'the two-interval index from the lattice', the mu = 1 indicator chain PASSES: fermionic two-interval MI extensive (c fit 0.5000, max |F| = 6e-5 at N = 512, falling with N -- the mu = 1 reference), the sector-summed orbifold prescription pays exactly one classical bit with the 16-layer seam carrier saturating Delta_2 = ln 2 to machine precision (1.1e-15 at N = 512), so [F : F_even] = 2 and mu_gauged = 4 = the v490 parity census (two independent lattice witnesses); complementary-pair budget <= ln 4 = ln mu(SO(16)_1) as a double-limit statement + the orbifold duality-failure witness D(E) != D(E'); condensation arithmetic anchored at both ends: det Cartan(D5) x det Cartan(A3) = 16, KLM/Longo-Rehren 16/4^2 = 4/2^2 = 1, Sigma d^2 = (4,4,1), theta_v = 1 exactly at nu = 16 (rivals != 1) => mu = 1 after condensation, preregistered KILL 'mu-offset != 0 after condensation' NOT triggered; controls: nu = 1 offset non-removable, trivial phase all zero, wrong sector sum loses the full ln 2; WP5d-beta DONE (v504, 2026-07-21 -- 'split + strong additivity from the lattice', the two remaining KLM legs: strong additivity ALGEBRAICALLY EXACT with the shared boundary Majorana (Even(A) v Even(B) = Even(A u B), GF2 spans full 64/64..1024/1024, matrix rank 32/32; disjoint exactly index 2 = the v501 ln-2 bit, localised at the split point), the entropic touching defect BOUNDED < ln 2 with the Ising 1/4-exponent approach (p = 0.2444 vs 2 Delta_mu = 1/4) while the preregistered U(1)/Dirac control DIVERGES ((1/2) ln Var Q, Klich-Levitov pinned, infinite index) -- bounded-vs-divergent is the lattice discriminator and the quotient net sits on the bounded side; split via the elliptic-nome sigma_k ladder (rate pi K(1-x)/K(x) to 1.3-2.0%, sigma_1 ~ x^{1/2}, trace norm summable) with EXACT orbifold inheritance (C -> -C machine identity + the Lambda^2 C compound -- Longo heredity); Pimsner-Popa E(a) - a/2 = PaP/2 identically, lambda = 1/2 = 1/[F:F_even] with exact integer attainment (16384 + 2048 monomials, 0 violations), lambda_E4 = 1/4 = 1/mu, index consistency exp(Delta) = 2 = 1/lambda_PP over two independent routes: WITH v501 ALL THREE KLM INGREDIENTS of complete rationality -- split, strong additivity, finite mu -- are witnessed on the lattice; the continuum uplift stays [O]: 'finite-group orbifolds of completely rational nets are completely rational' is XU'S THEOREM, cited not claimed -- the proof for the concrete seam quotient net and the interacting condensed (E8)_1 net remains WP5e/Costello-Li); WP5e: alpha stage (CFT-side prefactor + level pinning) executed via v502 (2026-07-21, CELEST.WP5E.ALPHA.01 -- the q^{-1/3} prefactor derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions) and k = 1 forced three independent ways (current condition h(J) = k = 1, conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0, c = 8 <=> 240(k-1) = 0, plus the WP5b singular-vector dial 31124 - 27000 = 4124), with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing); beta stage (equivariant twistor skeleton) executed via v505 (2026-07-21, CELEST.WP5E.BETA.01 -- the Atiyah-Bott/Lefschetz fixed-point skeleton of the inflow exact: denominators (2,4,2), characters (248,0,-8,0) two routes, per-sector Okubo with the RIGID 32 T3 twisted residual, index bridge f(m) = ch2(T_m) with glue defect -78 both routes, level dials k = 1 geometric (current count 240/0/0/0), and the E3 refutation a0 = graviton modulus O(2) NOT the axion O(-2) with the three sphere classes filling the three twisted axion slots; the preregistered kill 'inflow demands a level != 1' does NOT fire on the equivariant skeleton); gamma stage (the sphere-axion pairing check) executed via v508 (2026-07-21, CELEST.WP5E.GAMMA.01 -- an honest rigid NEGATIVE result with certificate: the W(D5) x W(A3)-invariant vertex spaces are exactly dim 2 (quadratics) and dim 5 (quartics), the product of any two invariant quadratics has zero T5/T3 content, so the exchange matrix is rank 2 with rank([M | A_fix]) = 3 and annihilator certificate (Phi_T5, Phi_T3, Phi_P)(A_fix) = (0, 32, 72) -- the three sphere axions CANNOT cancel the rigid 32 T3 residual by quadratic-vertex exchange, rule- and coupling-independently; the slot bijection is untouched and the preregistered level-kill still does not fire); WP5e: alpha (CFT side, v502) + beta (equivariant twistor skeleton, v505) + gamma (exchange no-go, v508) executed; the global BCOV quantisation stays [O] -- the exchange sub-branch is closed by v508; epsilon-2 executed via v509 (2026-07-21, CELEST.WP5E.EPS2.01, 'the CPS level-from-flux dial', verdict B): the one-level statement is now a THEOREM (lockstep from clock invariance -- one mu4 orbit of branch points, |Pi_j| = sqrt(2) t equal on all three spheres, det(A-1) = -4 no invariant flux vector; falsifier Z^4 - Z: 0/24 orderings lockstep), level = 1 carries the new sector-counter dial #prim((E8)_k) = 1 <=> k = 1 (affine marks: (1, 3, 5, 10, 15, 27) for k = 1..6), fluxes (64, 60, 64) with uniformly one quantum per charged current and the naive 'level = total flux' killed by the lockstep test itself; the PT/Z4 back-reaction stays [O]; delta-2 executed via v511 (2026-07-21, CELEST.WP5E.DELTA2.01, intermediate verdict, exact -- the full-tensor ledger: the v508 collapse is confirmed full-tensorially and arity-crossing by the innerness theorem (h in the g0 Cartan, all 15 non-neutral trilinear triples Hom = 0), but the su(4) d-symbol -- the unique symmetric trilinear on e8 -- opens the T3 channel (Q_dd = (1/60)(T3 - P3/4), Phi_T3(Q_dd) = 1/60 != 0), leaving the pairing obstructed only by the weaker certificate {Phi_T5, psi = Phi_P - Phi_T3/4}, psi(A_fix) = 64, and exactly solvable relaxed with c_d = 1920 = |W(D5)|); delta-2 executed via v511 -- the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T3 with the weaker certificate psi = 64; delta-1 since DECIDED NEGATIVE under the DERIVED measure by v518 (2026-07-22, CELEST.WP5E.DELTA1.01: the derived chiral measure -- blockwise SL(2,Z) covariance solved for, the mu4 multiplier obstruction a CHARACTER (koboundary defects (1,1,1) on all 15 pairs, lambda(gamma) = i^{2B + C/4} on Gamma_1(4)) cancelled exactly by the twisted fibre block f1 f3 = G, NOT by the three sphere axions -- fails all three preregistered testers under both derived solutions chi_4/chi_10, with no (N1, N2) rescue in the positive cone: a genuine KILL on the derived surface, in stated TENSION with the DECLARED completion reading of v516 which delivers the psi = 64 slice and cancels 32 T3 -- both are exact, and which of the two is the true BCOV measure is the named [O]); the pre-v518 state was: the strict holomorphic reading is refuted at the Z2 anchor, the contact term is the modular completion (Harvey-Moore tau-integral) with the forced leading (T5,T3) ratio 4:3 -- the named remaining targets are the tau-integral and the physical justification of the cubic GS term (c_d = 1920; the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513; the convention-stable [E] core is c_d = 32 x 60), plus the epsilon-1 O(-2) bulk-axion construction (success: partition function = E4/eta^8 including the q^{-1/3} prefactor FROM THE TWISTOR SIDE; kill: anomaly mismatch at the Costello-Li grade -- neither the CFT-side dials of v502 nor the equivariant skeleton of v505 trigger it). KILL TESTS (pre-registered): (K1) the equivariant structure could have contradicted 52+64+60+64 -- SURVIVED by WP1 AND WP2 (false-glue control: 3112/6720 violations show the test has teeth; WP2 deformation side: no shape modulus survived clock-invariance, no sector leak -- the clock-variant controls fail two independent ways each); (K2) one-loop associativity demanding extra fields TFPT lacks -> contract dies (Costello's axionic mechanism passes E8 at the algebra level; the TFPT field content is the open half); (K3) EVALUATED by WP3 (v495): does NOT fire -- exact alignment exists ((kappa/c3)^2 = 12 resp. 1/300); but its SCOPE IS DEMOTED on the look-elsewhere evidence (the alignment format passes 8/8 across Costello's whole list): compatibility, not e8-selective evidence; (K4) EVALUATED by WP4 (v496): fires only against the exact-block reading -- the sector arithmetic holds exactly, so the contract does NOT degrade to 'E8 is an admissible celestial gauge algebra'; WP4 lands at the boundary-limit reading (verdict B(ii), the SEAM.EQUIV.01 scaling-limit shape). HONEST SUCCESS ESTIMATE: full success (WP5e, constructive holography at the Costello-Li grade) low probability; the partial-findings expectation is now REALISED AND EXCEEDED: WP1-WP4 + WP5a-WP5d all executed with sharp structural results (verdicts B, B, B, B(ii); ALL FIVE constructive WP5 milestones landed -- null ideal, singular vector, GNS limit state, two-interval index, split + strong additivity (both WP5d stages) -- plus the WP5e alpha stage (CFT-side prefactor + level pinning, v502) AND the WP5e beta stage (equivariant twistor skeleton with the honest a0 refutation, v505) AND the WP5e gamma stage (the exchange no-go with certificate, v508) AND the WP5e epsilon-2 stage (the CPS level-from-flux dial with the lockstep theorem and the sector-counter dial, v509); AND the WP5e delta-2 stage (the full-tensor ledger with the innerness theorem and the cubic d-symbol channel, v511); WP5e proper, the global BCOV quantisation, is the single remaining open piece -- narrowed by v508/v509/v511 to the delta-1 tau-integral (since DECIDED by v518, CELEST.WP5E.DELTA1.01: kill under the derived measure, all three preregistered testers fail; the measure tension with v516 is the named [O]), the physical justification of the cubic GS term (c_d = 1920; the |W(D5)| reading look-elsewhere-loaded and convention-contingent -- v513) and the epsilon-1 bulk-axion construction (executed via v514) plus the PT/Z4 back-reaction (M1 executed via v515: the back-reacted Omega_N closed-form with (2 pi i)^2-integral lockstep flux vector and forced source charge 4 = |mu4|; M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01)) -- the clock^2 = deck correction, the Coxeter/Picard-Lefschetz monodromy, the non-selective GS alignment, the boundary-limit localisation of the character mismatch, the explicit singular vector with its level dial 2(1-k), the limit state with the ideal in its GNS kernel, the entropically measured mu-chain 16 -> 4 -> 1, and the lattice KLM completeness witness (split + strong additivity + finite mu, v501/v504). SEAM.EQUIV.01 stays [O] throughout; nothing here moves it.
Recorded status: Open (research contract; WP1-WP5d executed [E]/[C]/[L] via v492/v493/v495/v496/v497/v498/v500/v501/v504 -- WP5d: alpha (mu-index) + beta (split + strong additivity) executed via v501/v504, all three KLM ingredients witnessed on the lattice, continuum uplift stays [O] (Xu's theorem cited); WP5e alpha stage [E] via v502 + beta stage [E] via v505 (equivariant twistor skeleton; kill not fired) + gamma stage [E] via v508 (exchange no-go with certificate (0, 32, 72); negative result, kill not fired) + epsilon-2 stage [E] via v509 (level-from-flux: the one-level statement is now a THEOREM of clock invariance, the sector-counter dial #prim((E8)_k) = 1 <=> k = 1 pins the level; the PT/Z4 back-reaction stays [O]); + delta-2 stage [E] via v511 (the full-tensor ledger: collapse confirmed by innerness, the cubic d-symbol channel opens T3 with the weaker certificate psi = 64, relaxed solution c_d = 1920 = |W(D5)|); + epsilon-1 stage [E] via v514 (epsilon-1 executed via v514: the O(-2) bulk-axion slot is a construction (equivariant Penrose ledger), lambda-tilde = 6 pinned by three exact ledgers (Okubo / measure cancellation excluding 3 and 12 / flux), and the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from geometry -- the quantised BCOV coefficient and the twisted channels stay [O] (M1-M3 preregistered)); WP5e-M1 DONE (v515: M1 executed via v515: the back-reacted Omega_N is closed-form on the A3 twistor family, all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector and clock covariance, and the lens geometry FORCES the source charge 4 = |mu4|; honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden family)); WP5e-M2+M3 DONE (v516/v517: M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01); the global BCOV quantisation [O] (single remaining milestone -- the exchange sub-branch closed by v508, the epsilon-2 flux dial executed by v509, the delta-2 tensor ledger executed by v511, the epsilon-1 axion slot executed by v514, the M1-M3 back-reaction milestones executed by v515/v516/v517 (the v514 fence fully worked off); named remaining target: the BCOV MEASURE QUESTION -- the delta-1 chain is DECIDED by v518 (CELEST.WP5E.DELTA1.01: kill under the derived measure, all three testers fail; tension with v516 stated), so what remains is which of the two exact measures (the declared v516 completion reading vs the v518 derived chiral measure) is the true BCOV measure; the cubic GS term stays dissolved (psi = 64 delivered by v516, no branch fires under the derived measure, v518)))
Dependencies, status detail and source
Canonical status: Open (research contract): seam = glue-equivariant Z4-orbifold sector of the CCA on the A3 ALE; clock^2 = deck (Z8 spin bridge); twistorial bulk = self-dual sector only; WP1-WP4 + WP5a-WP5d executed (verdicts B, B, B, B(ii); WP5a: null ideal derived, 27000 = h_vee^3, quotient = sector sum = 4124, stabilisation threshold w = n+1; WP5b: the singular vector explicit, J^a_1|s> = 0 on all 248 generators with case tally 190/57/1, level dial 2(1-k), j(theta) = 1, clock phase i^2 = -1), WP5c DONE (v500: GNS limit state omega_inf with the null ideal in its kernel -- level-2 rank 4124, kernel 27000 = V(2 theta) weight by weight, clock split (1036,1024,1040,1024), |s> = the GNS zero vector, CCR necessity of the family), WP5d-alpha DONE (v501: two-interval index -- 16-layer ln 2 plateau at 1e-15, [F : F_even] = 2, mu_gauged = 4 = v490, KLM chain 16 -> 4 -> 1 anchored at both measured ends, theta_v = 1 at nu = 16), WP5e-alpha DONE (v502: q^{-1/3} prefactor = exact mu4 vacuum-energy bookkeeping (shift orbifold, discriminant form = Casimir two ways) + k = 1 forced three ways, glue-h integrality honestly demoted), WP5e-beta DONE (v505: the equivariant anomaly ledger exact -- AB characters (248,0,-8,0), per-sector Okubo with the rigid 32 T3 residual, index bridge f(m) = ch2(T_m) with defect -78 two routes, geometric k = 1 dials, a0 refuted as GS axion (graviton slot O(2)), the three sphere classes = the three twisted axion slots; kill not fired), WP5d-beta DONE (v504: split + strong additivity from the lattice -- Even(A) v Even(B) = Even(AuB) EXACT with the shared boundary point (disjoint index exactly 2 = the v501 ln-2 bit), bounded-vs-divergent entropic discriminator (Z2 deficit < ln 2 with the Ising 1/4-exponent approach vs U(1) divergence, Klich-Levitov pinned), elliptic-nome split ladder with exact orbifold inheritance (C -> -C + Lambda^2 C, Longo heredity), Pimsner-Popa lambda = 1/2 = 1/[F:F_even] and lambda_E4 = 1/4 = 1/mu with integer attainment, index consistency exp(Delta) = 2 = 1/lambda_PP two routes -- all three KLM ingredients witnessed on the lattice; continuum uplift stays [O], Xu's theorem cited), WP5e-gamma DONE (v508: the sphere-axion pairing check -- exchange no-go with certificate: invariant vertex spaces exactly dim 2/5 by Weyl nullspace, product theorem kills every exchange image in the T3 direction, rank([M | A_fix]) = 3 with certificate (Phi_T5, Phi_T3, Phi_P) = (0, 32, 72), naturalness dissolves scale-independently, SO(16)/D8 controls; slot bijection untouched, level-kill still not fired), WP5e-epsilon-2 DONE (v509: the CPS level-from-flux dial, verdict B -- the one-level statement is now a THEOREM (lockstep from clock invariance, falsifier Z^4 - Z: 0/24), level = 1 carries the new sector-counter dial #prim((E8)_k) = 1 <=> k = 1, fluxes (64, 60, 64) with one quantum per current, the naive 'level = total flux' killed; the PT/Z4 back-reaction stays [O]), WP5e-delta-2 DONE (v511: the full-tensor ledger -- delta-2 executed via v511 -- the full-tensor collapse is confirmed (innerness), but the cubic d-symbol channel opens T3 with the weaker certificate psi = 64; delta-1 since DECIDED NEGATIVE under the DERIVED measure by v518 (2026-07-22, CELEST.WP5E.DELTA1.01: the derived chiral measure -- blockwise SL(2,Z) covariance solved for, the mu4 multiplier obstruction a CHARACTER (koboundary defects (1,1,1) on all 15 pairs, lambda(gamma) = i^{2B + C/4} on Gamma_1(4)) cancelled exactly by the twisted fibre block f1 f3 = G, NOT by the three sphere axions -- fails all three preregistered testers under both derived solutions chi_4/chi_10, with no (N1, N2) rescue in the positive cone: a genuine KILL on the derived surface, in stated TENSION with the DECLARED completion reading of v516 which delivers the psi = 64 slice and cancels 32 T3 -- both are exact, and which of the two is the true BCOV measure is the named [O]); the pre-v518 state was: the strict holomorphic reading is refuted at the Z2 anchor, the contact term is the modular completion (Harvey-Moore tau-integral) with the forced leading (T5,T3) ratio 4:3 -- the named remaining targets are the tau-integral and the physical justification of the cubic GS term (c_d = 1920; the 1920 = |W(D5)| reading is look-elsewhere-loaded (11/924 vs. 8/924 for the control target 1800) and convention-contingent -- v513)), WP5e-epsilon-1 DONE (v514: epsilon-1 executed via v514: the O(-2) bulk-axion slot is a construction (equivariant Penrose ledger), lambda-tilde = 6 pinned by three exact ledgers (Okubo / measure cancellation excluding 3 and 12 / flux), and the GH/A3 back-reaction step re-derives the v493 family and the Coxeter clock from geometry -- the quantised BCOV coefficient and the twisted channels stay [O] (M1-M3 preregistered)); WP5e-M1 DONE (v515: M1 executed via v515: the back-reacted Omega_N is closed-form on the A3 twistor family, all S3/Z4 periods are (2 pi i)^2-integral with the lockstep flux vector and clock covariance, and the lens geometry FORCES the source charge 4 = |mu4|; honest fence: integrality alone does not discriminate -- the discriminator is the lockstep phase structure (0/24 on the forbidden family)), WP5e-M2+M3 DONE (v516/v517: M2+M3 executed via v516/v517 -- the v514 fence M1-M3 is fully worked off: the twisted KS measure (declared completion reading, ch2-weighted) cancels the 32 T3 residual and delivers the psi = 64 slice without the cubic d-channel, and the a0 uplift couples to the centre count on four scales; what stays [O]: the BCOV measure question -- the delta-1 chain itself is DECIDED by v518 (kill under the derived measure, tension with v516 stated) -- and SEAM.EQUIV.TWISTOR.01), the global BCOV quantisation [O] (the single remaining milestone: Costello-Li uplift -- the exchange sub-branch closed by v508, the flux dial executed by v509, the tensor ledger executed by v511, the epsilon-1 axion slot executed by v514, the M1-M3 back-reaction milestones executed by v515/v516/v517; the named remaining target is the BCOV measure question (the delta-1 chain is DECIDED by v518: kill under the derived measure, tension with v516 stated; the cubic GS term stays dissolved)); kill tests evaluated: K1 survived, K3 not fired (scope demoted -- alignment non-selective, compatibility not evidence), K4 lands at the boundary-limit reading; SEAM.EQUIV.01 untouched [O]
Dependencies: CELEST.WP1.01; CELEST.WP2.01; CELEST.WP3.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; CELEST.WP5DB.01; CELEST.WP5E.ALPHA.01; CELEST.WP5E.BETA.01; CELEST.WP5E.GAMMA.01; CELEST.WP5E.EPS2.01; CELEST.WP5E.DELTA2.01; CELEST.WP5E.EPS1.01; CELEST.WP5E.M1.01; CELEST.WP5E.M2.01; CELEST.WP5E.M3.01; CELEST.WP5E.DELTA1.01; SEAM.EQUIV.01; CONTRACT.G.01; GATE.METRIC.01; v92; v125; v128; v168; v180; v214; v492; v493; v495; v496; v497; v498; v500; v501; v502; v504; v505; v508; v509; v511; v514; v515; v516; v517; v518
Scripts: v492_celestial_z4_orbifold.py
Location: tfpt_research_contracts
Original CSV, line 556Original ledger status
CELEST.WP2.01
WP2 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-20; CELEST.SEAM.01 work package 2, verdict B; companion of CELEST.WP1.01/v492): the clock-invariant deformation XY = Z^4 + a0 of the A3 seam orbifold -- geometry, period/monodromy household, and the Bittleston-Homans-Sharma deformed-algebra pattern (arXiv:2305.09451, JHEP 09 (2023) 008; the Z2/Eguchi-Hanson case, fibre XY - Z^2 = -c^2) transferred to k = 4. Sympy exact throughout (47 checks). [E] GEOMETRY: P(iZ) = P(Z) forces a3 = a2 = a1 = 0 SHARPLY and two-sided (prod_k (Z - i^k w) = Z^4 - w^4: mu4-stable root set <=> clock-invariant quartic); XY = Z^4 + a0 smooth iff a0 != 0; disc = 256 a0^3; branch points Z^4 = -a0 = ONE free mu4 orbit in every fibre; cross-ratio 2 / j = 1728 for ALL a0 (binary-quartic invariants I = 12 a0, J = 0 IDENTICALLY) => a0 is a PURE SCALE, the tau = i pillowcase (v168/v214) is FROZEN -- no shape modulus survives clock-invariance. [E] PERIODS / MONODROMY: Pi = t(i-1)*(1, i, i^2) -- all 3 vanishing spheres carry the SAME holomorphic volume sqrt(2) t (lockstep, no relative modulus); the clock acts on H2 as a COXETER ELEMENT of W(A3) (char x^3+x^2+x+1, eigenvalues {i,-1,-i} = the A3 exponents 1,2,3; order = Coxeter number h(A3) = 4 = |mu4| = N_fam + 1); NO invariant cycle (det(A-1) = -4); the surviving deformation direction is the weight-1 chi_1-Fourier diagonal (1, i, i^2), NOT the naive invariant diagonal; the clock IS the Picard-Lefschetz monodromy of the family (a0 -> e^{2 pi i} a0 = the 4-cycle); versal weights (a3,a2,a1,a0) = (1,2,3,0) = the REGULAR mu4 representation, invariant line = a0 alone. [E] DEFORMED ALGEBRA (BHS Z2 -> Z4): mode dictionary {w[m,n]: m = n mod 4} <-> {X^p Y^q Z^r: r <= 3}; flat deformation (counting level); bracket {f,g} = -4*Nambu(XY - Z^4 - a0) ANCHORED at the a0 = 0 orbifold w-sector ({X,Y} = 16 Z^3, {Y,Z} = -4Y, {Z,X} = -4X), Jacobi + Casimir exact, uniform clock-degree -1; S-algebra corrections exactly LINEAR in a0 at the Z4 wrap r+r' >= 4, conserving the mu4 grade -- NO sector leak, so the v492 glue-equivariant SDYM(E8) sector deforms consistently; a0 <-> the BHS c^2 slot (weight 0, the order-c^2 splitting-function correction slot); matrix factorisations (McKay modules) deform along every 1+3 and 2+2 orbit splitting; the clock fixes exactly the antipodal 2+2 sector = the EH/Z2 middle node (4 = 2 x 2). [E] NEGATIVE CONTROLS (doubly sharp): a1 Z => j = 0 (hexagonal CM point, maximally non-square); pure a2 Z^2 stays SINGULAR (A1 node); a2 != 0 => j = 1556068/81 != 1728 (exact instance, cross-ratio and binary-quartic routes agree); grading leak explicit (Z^2 * Z^2 picks up grades {0,1,2} in the variant ring). [C] THREE NAMED IDENTIFICATIONS (why verdict B, not A): (1) the fibrewise -4*Nambu bracket is THE k = 4 CCA bracket (BHS prove k = 2; anchored here at a0 = 0, twistorial derivation not redone); (2) period = root difference (standard A-series residue normalisation); (3) the seam-scale reading of a0 presupposes the WP1 clock-deck bridge (clock^2 = deck, Z8, 8 = 2|mu4|). [C] v216 PRECISION (typed, NO marker move): GIVEN the order-4 clock, the clock-invariant deformation freezes the (2,2,2,2) modulus to the square AUTOMATICALLY (only the scale a0 stays free) -- a RELOCATION of the same order-4 carrier input to the celestial side (the clock IS the input), not a new derivation; v216 keeps its [O]-residual. [O] OPEN: full deformed twisted-sector OPEs; the W(mu)-type scaling-limit identification for k = 4 (BHS announce the ADE case); the spacetime splitting-function check; WP3 (GS coefficient alignment), WP4 (character vs S-algebra grading), WP5 (constructive holography); SEAM.EQUIV.01 (continuum (E8)_1 net on the seam) stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact deformation geometry/period/algebra arithmetic, sympy, negative controls) + [C] (three named identifications, verdict B; v216 precision typed) + [O] (algebra/contract programme; SEAM.EQUIV.01 and v216 unchanged)
Dependencies, status detail and source
Canonical status: WP2 [E]/[C]/[O]: clock-invariance selects XY = Z^4 + a0 sharply; a0 = pure seam scale, tau = i pillowcase frozen (j = 1728 for all a0); clock on H2 = Coxeter element of W(A3) = Picard-Lefschetz monodromy; surviving direction = chi_1-Fourier diagonal, spheres in lockstep; BHS Z2 -> Z4: -4*Nambu bracket anchored at the orbifold, corrections linear in a0, grade-conserving (no sector leak), a0 <-> c^2 slot; negative controls j = 0 / j != 1728 / grading leak; verdict B; SEAM.EQUIV.01 and v216 [O] untouched
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v168; v179; v180; v214; v216; v492
Scripts: v493_celestial_wp2_clock_deformation.py
Location: tfpt_research_contracts
Original CSV, line 557Original ledger status
CELEST.WP3.01
WP3 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 3, verdict B; the pre-registered 'alignment test only'): the Green-Schwarz axion coefficient lambda_g of celestial/twistorial SDYM (Costello, arXiv:2111.08879, eq. 1.2.4: kappa = lambda_g/(8 pi sqrt 3)) against the TFPT seam constant c3 = 1/(8 pi). Exact arithmetic throughout (Fraction/sympy, 25 checks). [E] OKUBO IDENTITY DERIVED, NOT QUOTED: Sum_alpha <alpha,x>^4 = [5/(2(dim g+2))] (Sum_alpha <alpha,x>^2)^2 as an exact polynomial identity for ALL 8 algebras on Costello's list (sl2, sl3, g2, so8, f4, e6, e7, e8; e6/e7 carved out of the v492 E8 root set) -- 'no independent quartic Casimir'; NEGATIVE CONTROL: sl4 (which HAS one) fails the proportionality (5/32 on s0^4 vs 3/32 on s0^2 s2^2) -- the list is sharp. [E] Killing normalisation Sum <alpha,x>^2 = 2 h_vee <x,x> exact (h_vee = 2,3,4,6,9,12,18,30); Deligne dimension formula dim g = 2(5h_vee-6)(h_vee+1)/(h_vee+6); CLOSED FORM lambda~^2 = 10 h_vee^2/(dim+2) = h_vee + 6 EXACTLY (8,9,10,12,15,18,24,36) -- fixes Costello-Paquette eq. (1.5); fundamental Dynkin indices i_R = (1,1,2,2,6,6,12,60) from explicit weight systems. E8: lambda~ = 6 (unit-trace) and lambda_fund = 1/10 (adjoint-trace) EXACT. Cross-checks: Costello's sl2 = 8, sl3 = 9 reproduced; HIS PRINTED lambda^2(so8) = 3/2 is a factor-2 slip (his own weight content gives Sum w^4 = 192, not 96: lambda^2 = 3, Okubo-consistent 12/4). [E] ALIGNMENT: (kappa/c3)^2 = lambda^2/3 = 12 = |mu4| N_fam (unit-trace) resp. 1/300 (adjoint-trace), both EXACT anchor rationals; kappa/c3 ITSELF is IRRATIONAL (2 sqrt 3 resp. 1/(10 sqrt 3)) -- the pre-registered 'lambda/sqrt3 rational?' reading FAILS, only the SQUARED alignment holds; anchor decomposition h_vee = 30 = |Z2| N_fam g_car (all three TFPT atoms once, squarefree), phi(30) = 8 = rank(E8), dim+2 = 250 = 2 * 5^3, lambda~ = 6 = |Z2| N_fam, lambda_fund = 1/10 = 1/(|Z2| g_car). [E] LOOK-ELSEWHERE (the HARD FENCE -- alignment != selection): the alignment test AS SUCH passes 8/8 (ZERO selective power for e8), h_vee {2,3,5}-smoothness 8/8, lambda~-integrality 2/8 (sl3 + e8), lambda_fund-rationality the same 2/8, phi(h_vee) = rank 4/8, phi(h) = rank 5/8; the ONLY e8-selective single test is g_car = 5 | h_vee (1/8); the conjunction that isolates e8 (1/8) was assembled POST HOC from correlated sub-tests -- it quantifies where e8 sits, it is NOT a preregistered discriminator. [E] KILL-TEST K3 EVALUATED: does NOT fire (exact alignment EXISTS) but its scope is DEMOTED -- the c3-connection of the celestial route survives as convention-level compatibility + genuine lambda-arithmetic, NOT as e8-selective evidence. [C] TYPED: the 8 pi in Costello's (1.2.4) is the 2 pi convention of the anomaly polynomial (his A.0.10 constant descended to spacetime), NOT an independent occurrence of the seam constant; the sqrt 3 is the anomaly's own (sqrt 48 = 4 sqrt 3 from the twistor box diagram, 48 = 2*4!); WP3 is an alignment yes/no ONLY -- it can never derive c3 from celestial data (contract wording, honoured). [O] WP5 (constructive holography) and SEAM.EQUIV.01 stay [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact Okubo/Killing/lambda arithmetic for all 8 algebras, Fraction/sympy, sl4 negative control, look-elsewhere battery) + [C] (the convention typing of the 8 pi and sqrt 3; the alignment reading with the look-elsewhere caveat, verdict B) + [O] (celestial programme WP5; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP3 [E]/[C]/[O]: Okubo coefficient 5/(2(dim+2)) derived for all 8 algebras (sl4 control); lambda~^2 = h_vee + 6 closed form; lambda_e8 = 6 resp. 1/10 exact; (kappa/c3)^2 = 12 = |mu4| N_fam resp. 1/300 exact, kappa/c3 irrational; so8 slip flagged; look-elsewhere: alignment 8/8 (zero selectivity), g_car | h_vee 1/8 (e8 only), joint test post hoc; K3 not fired, scope demoted -- compatibility, not e8-selective evidence; verdict B; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v92; v125; v128; v492; v493
Scripts: v495_celestial_wp3_gs_alignment.py
Location: tfpt_research_contracts
Original CSV, line 558Original ledger status
CELEST.WP4.01
WP4 OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 4, verdict B(ii); the 'type mismatch, head-on'): the (E8)_1 character E4/eta^8 against the glue-equivariant E8[C^2] S-algebra. Exact integer/Fraction/sympy arithmetic (25 checks). [E] BASE: v492 replication (240 = 52+64+60+64, dims with Cartan (60,64,60,64)); equivariant tower D[d] = 60(d+1) even / 64(d+1) odd; cumulative generator count QUADRATIC -- 31 s^2 + 92 s + 60 (even) / 31 s^2 + 94 s + 63 (odd) exactly, leading 31 = k + h_vee (arithmetic note); [C] CP Table 1 (arXiv:2201.02595): J[m,n] carries 2d spin 1 - (m+n)/2, UNBOUNDED BELOW along the tower => no rational VOA (L0 bounded below) contains the towers as graded pieces. [E] EQUIVARIANT HILBERT SERIES: chi_sp(t) = (60 + 128t + 60t^2)/(1-t^2)^2 (palindromic numerator, value 248 at t = 1); ALL mu4 collapses w in {1,-1,i,-i} and the SU(2)_+ primary slice (60+64t)/(1-t^2) fail to start (1, 248, ...) -- the 'row sum' route fails at the first coefficient. [E] CHARACTER SIDE: chi = E4/eta^8 = (1, 248, 4124, 34752, 213126, 1057504, 4530744) exact through q^6; machinery cross-check q j(tau) = E4^3/eta^24 = (1, 744, 196884, 21493760); theta split (shell 1 = (52,64,60,64)), sector currents (60,64,60,64), glue-diagonal h = (0,1,1,1) INTEGER (v125 isotropy = locality); Sugawara c = 248/31 = 8 = rank = 2|mu4| = g_car + N_fam. [E] BRIDGES: zero-mode slice D[0] = 60 = the C0-sector current count EXACTLY; the single-line jet slice z1^d cycles (60,64,60,64) sector by sector; BOUNDARY/PERIOD LIMIT: one full mu4 period of loop energies carries dim g_(n mod 4) = (64,60,64,60), sum = 248 EXACTLY -- the single-particle level of the 248q stage; 248 is NOT a jet-tower dimension (d <= 100) -- only the period collapse produces it; BUT the loop Fock at integer level 1 counts 897266 >> 248: the rational truncation (null ideal) is NOT in the jet/loop algebra -- it must come from a limit. [E] MISMATCH LOCALISED THREE WAYS: (a) GRADING (spin unbounded below; 248 no tower dimension); (b) GROWTH: jet Fock f = (1, 128, 8436, 381056) vs chi = (1, 248, 4124, 34752) -- TWO-SIDED failure (f1 < chi1 undercount, f2 > chi2 overcount), f_n/chi_n strictly increasing n = 1..12 (exact cross-multiplication); Meinardus poles s = 2 vs s = 1 => growth exponents n^{2/3} vs n^{1/2}, asymptotically DISJOINT counting types; (c) TRUNCATION: free-current level-2 count 31124 = 248 + Sym^2(248); the character keeps 4124 = 1 + 248 + 3875; the deficit is EXACTLY 27000 = 30^3 = h_vee^3 -- the (E8)_1 null ideal deletes the 27000 in Sym^2(248) = 1 + 3875 + 27000 ([C] standard) with NO analogue in the jet algebra (level 1 agrees: 248 = 248). [E] KILL-TEST K4 EVALUATED, three-way: (i) exact realisation NO; (ii) boundary-limit realisation PARTIAL YES -- current stratum, period sum 248, integer glue weights and theta split hold exactly; the missing piece is precisely the rational truncation, which must come from a scaling/compactification limit -- the SAME structure as the MMST scaling-limit route of SEAM.EQUIV.01 (v336/v449: the rational net exists in the LIMIT of the lattice collar, not in the pre-limit algebra); (iii) the hard mismatch is exactly located; K4 does NOT degrade the contract to 'E8 is an admissible celestial gauge algebra' (the sector arithmetic holds exactly). [C] VERDICT B(ii): the (E8)_1 character is a BOUNDARY/LIMIT SHADOW of the equivariant S-algebra, not a conformal block of it in the bulk (jet) grading -- WP4 sharpens into a statement of SEAM.EQUIV.01 type (rational structure emerges only in a limit). [O] the constructive boundary limit itself (which limit, which topology, convergence) stays OPEN and belongs to WP5/SEAM.EQUIV.01; SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact counting/character/theta arithmetic, integer/Fraction/sympy; mismatch localisation with exact cross-multiplication) + [C] (literature gradings CP Table 1 + Sym^2(248) decomposition; the boundary-limit reading, verdict B(ii)) + [O] (the constructive limit -> WP5; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP4 [E]/[C]/[O]: the (E8)_1 character is NOT a conformal block of the E8[C^2] S-algebra in its own jet grading -- mismatch localised three ways (spin unbounded; growth n^{2/3} vs n^{1/2}; level-2 null ideal 27000 = h_vee^3 absent from the jets); BUT the boundary/period reading holds exactly at the current stratum (zero modes 60, jet slice (60,64,60,64), mu4 period sum 248, glue weights (0,1,1,1) integer) -- the MMST scaling-limit shape of SEAM.EQUIV.01; K4 fires only against the exact reading; verdict B(ii); constructive limit -> WP5; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492
Scripts: v496_celestial_wp4_character_shadow.py
Location: tfpt_research_contracts
Original CSV, line 559Original ledger status
COSMO.RUNNING.01
Spectral-index running of the Starobinsky branch, PROMOTED from the 2026-07-20 scorecard prediction-of-record row (same N_star_reheating group as n_s/r/A_s; never an independent hit). [E] LO identities (sympy exact): alpha_s = dn_s/dln k = -2/N_star^2 = -r/6, beta_s = -4/N_star^3 (NOT -8/N^3); LO band N_star in [50,60]: [-8.0e-4, -5.6e-4]. [E] exact slow roll on the Einstein-frame Starobinsky potential (third-order Lyth-Riotto + finite-difference cross-check <1%): alpha_s(51.4) = -7.09e-4, beta_s(51.4) = -2.70e-5; exact band edges -7.48e-4 / -5.23e-4. [C] data confrontation: Planck 2018 TTTEEE+lowE+lensing (-0.0045+/-0.0067) -> +0.57 sigma; record leg P-ACT-LB (+0.0062+/-0.0052) -> -1.33 sigma (watch channel: the central value is POSITIVE); beta_s (Planck run+runrun, 0.010+/-0.013) -> -0.77 sigma. Forecast: LiteBIRD+CMB-S4 sigma(alpha_s) ~ 4.9e-3 (0.14 sigma); CMB-S4+DESI+Euclid statistical ~3.3e-4 (2.2 sigma, systematics-limited). KILL: robust alpha_s < -5e-3 or > +3e-3 at >=5 sigma; any robust POSITIVE running kills the plateau branch. FENCE: N_star stays an external reheating input (GATE.NSTAR.01); the frozen band [50,60] stays the surface of record; no gate closes, no marker moves
Recorded status: Identity [E] (LO relations + exact slow roll) + Physical [C] (data confrontation; N_star input)
Dependencies, status detail and source
Canonical status: [E] LO identities + exact-SR values; [C] pulls +0.57 sigma (Planck) / -1.33 sigma (P-ACT-LB record leg); kill = robust positive running; band [50,60] stays the record
Dependencies: COSMO.INF.01; COSMO.NSTAR.01; GATE.NSTAR.01
Scripts: v494_cosmo_running_mu_record.py
Location: tfpt_1
Original CSV, line 560Original ledger status
COSMO.MUDIST.01
CMB mu-distortion as the A_s/N_star BRANCH DISCRIMINATOR, PROMOTED from the 2026-07-20 scorecard prediction-of-record row (data-limited; shares the Nstar_branch alternative group with the two A_s readings -- one branch question, several readings). [C] Chluba-Erickcek-Ben-Dayan k-space window calibrated +21% on the LCDM Planck-2018 anchor to the literature mu_LCDM ~ 2.0e-8: sharp branch N_star = 51.4 (A_s = 1.76e-9) mu = 1.6e-8; profiled record N_star = 56.1 (A_s = 2.10e-9) mu = 1.94e-8 (~2.0e-8); band N_star in [50,60]: 1.5e-8 .. 2.3e-8; branch split Delta mu = 3.0e-9 (ratio 1.23) => >=3 sigma branch decision at sigma(mu) = 1e-9 (Voyage-2050 class). REACH: the PIXIE baseline 95% limit 9e-8 does NOT reach the band (sharp = 0.18 x limit); FIRAS 9e-5 is 4 dex above -- data-limited until a PIXIE+/Voyage-2050-class mission. [E] COMB BLINDNESS (honest S15 typing): the frozen recovery log-comb (omega = 2 pi/ln((3/2)^6) = 2.5827, eps = 0.0173) is STRUCTURALLY invisible in the mu window -- attenuation |F(omega)| = 0.033, delta mu = 7.4e-12 = 0.007 sigma even at Voyage-2050; the theory itself predicts where it cannot be seen, so NO comb-search row is opened. KILL: robust mu < 0.9e-8 or > 4e-8. FENCE: N_star stays an external input (GATE.NSTAR.01); the frozen band [50,60] stays the surface of record; no gate closes, no marker moves
Recorded status: Physical [C] (Chluba-window bridge, data-limited) + Identity [E] (comb-blindness attenuation)
Dependencies, status detail and source
Canonical status: [C] mu band 1.5-2.3e-8; branch split 3.0e-9 = 3 sigma at Voyage-2050; PIXIE baseline does not reach; [E] frozen comb structurally blind (0.007 sigma); kill = robust mu outside [0.9, 4]e-8
Dependencies: COSMO.INF.01; COSMO.NSTAR.01; GATE.NSTAR.01
Scripts: v494_cosmo_running_mu_record.py
Location: tfpt_2
Original CSV, line 561Original ledger status
CELEST.WP5A.01
WP5a OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5a, the first constructive milestone of WP5 -- 'the null ideal from the limit'; the v496 boundary-limit shadow made precise): exact integer/Fraction arithmetic throughout (34 checks), no floats. [E] STABILISATION THEOREM: a one-parameter family chi_w of graded Fock characters on the CHIRAL jet generators (loop energy m = p-q >= 1, radial number r = min(p,q); the m = 0, r >= 1 invariant-ring tower carries g_0), generator energy E_w = m + w*r in quarter units (u = q^{1/4}). (i) w = 2 IS the chiral jet grading (E_2 = p + q identically; generator counts 64,120,128,180,192,240,256,300 per degree); (ii) the u^n coefficient of chi_w equals the quarter-moded loop Fock for ALL w >= n+1 and is STRICTLY larger for every w <= n (verified n <= 8, w <= 10) -- the v496 'boundary-limit shadow' is the precise coefficientwise limit epsilon = 1/w -> 0 of a family CONTAINING the jet grading, with explicit threshold w = n+1; (iii) the limit does NOT produce the character (897266 >> 248 at integer level 1): limit and quotient are two separate, honestly typed steps; convention clarified: 897266 = quarter-moded loop Fock at u^4 = INTEGER level 1 (v496 S4.5). [E] NULL IDEAL DERIVED FROM ROOT DATA (not cited): Freudenthal multiplicity machinery (unit-tested on the adjoint: zero-weight mult 8, dim 248 by Weyl = orbit sum), dominant-weight tally of Sym^2(248) (30876 pairs, zero-weight mult 156) and generic character peeling DERIVE Sym^2(248) = V(2 theta) + V(omega) + V(0) = 27000 + 3875 + 1 (peeling residual exactly zero; independent Weyl-orbit cross-check); 27000 = 30^3 = h_vee^3 EXACTLY; C2 = (124, 96, 0) with 124 = C2(27000), Dynkin indices 6750/750 integers; level-1 integrability <2 theta,theta_vee> = 4 > 1 and <omega,theta_vee> = 2 > 1 => the only level-1 primary is the vacuum ([C] Kac); the maximal submodule of the level-1 vacuum Fock is generated by the level-2 singular vector of weight 2 theta ([C] Kac, used only through its counting consequence): level-2 quotient 31124 - 27000 = 4124 = 1 + 248 + 3875 = chi_2 EXACTLY. [E] TWO ROUTES, ONE NUMBER: the mu4 theta-split sector characters Theta_Cj/eta^8 (v492 S4 machinery, the boundary/twisted presentation) summed at q^2 = (1036,1024,1040,1024) give the SAME 4124 -- the untwisted null-ideal quotient and the mu4-sector lattice sum count the same level-2 space; at q^3 the sum is 34752 = chi_3; ideal bookkeeping beyond level 2 typed [C] (ideal series (0,0,27000,2600000,...) nonnegative, strictly increasing through q^12). [E] NEGATIVE CONTROLS: (a) SO(16)_1 through the SAME pipeline (D8 frame): Sym^2(120) = 5304 + 1820 + 135 + 1 derived (FOUR components vs E8's three); 5304 != h_vee(so16)^3 = 14^3 = 2744 -- the h_vee^3 identity is NOT generic; quotient 7380 - 5304 = 2076 = Theta_D8/eta^8 at q^2 (independent lattice route: the pipeline is the correct general level-1 vacuum recipe, validated on a second algebra); BUT the D8 block weights are (0, 1/2, 1, 1) -- h = 1/2 non-integer, the four blocks can NOT fuse into one local character (four-block theory) vs E8's integer (0,1,1,1) fusing into E4/eta^8: the procedure is not trivially projective; (b) false periodisation: period sums over P = 1, 2, 8 quarter-steps give 64, 124, 496 -- only P = 4 = |mu4| gives 248; (c) swapped glue (60/64 exchanged): the zero-mode anchor breaks (D[0] = 64 != 60 = lattice C0 currents). [C] NAMED IDENTIFICATIONS: 'radial decoupling = MMST collar thickness -> 0' (the v336/v449 lattice-collar route has the SAME two-step shape: pre-limit overcounting algebra, rational net only in the limit) is a named identification, not a computed isomorphism; the Kac singular-vector statement enters only through its counting consequence. [O] HONEST LIMIT (= the rest of WP5): the limit does NOT generate the truncation -- the singular vector as an OPERATOR in the limit net stays open (WP5b); the GNS limit state (kernel containing the ideal) and local normality / Haag duality of the limit net stay open (WP5c/WP5d); convergence topology beyond coefficientwise stays open; the celestial route now has the SAME two-step shape as MMST (limit + maximal ideal), with the ideal quantitatively fixed. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact stabilisation/derivation/control arithmetic, integer/Fraction; Freudenthal + Weyl + peeling machinery unit-tested) + [C] (the named identifications: radial decoupling = collar limit; Kac singular vector via counting consequence) + [O] (the operator question WP5b-e; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP5a [E]/[C]/[O]: the boundary limit is a precise coefficientwise limit -- chi_w family (E_w = m + w r, quarter units) contains the chiral jet grading at w = 2 and stabilises to the quarter-moded loop Fock at threshold w = n+1 (exact, n <= 8, w <= 10); the limit does NOT produce the truncation (897266 >> 248) -- but the ideal SIZE is DERIVED from root data (Sym^2(248) = 27000 + 3875 + 1, 27000 = h_vee^3, Freudenthal + Weyl + peeling) and the quotient 31124 - 27000 = 4124 = the independent mu4 theta-split sector sum; SO(16)_1 negative control (four blocks, 5304 != 14^3, h = 1/2 breaks fusion); only P = |mu4| = 4 gives 248; open: singular vector as operator (WP5b), GNS limit state (WP5c), local normality/Haag duality (WP5d), Costello-Li uplift (WP5e); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; SEAM.EQUIV.01; v125; v128; v336; v377; v449; v492; v496
Scripts: v497_celestial_wp5a_null_ideal.py
Location: tfpt_research_contracts
Original CSV, line 562Original ledger status
CELEST.WP5B.01
WP5b OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5b, the second constructive milestone of WP5 -- 'the singular vector as an operator'; WP5a/v497 fixed the ideal's size and location, WP5b constructs the deleting object): exact integer/Fraction arithmetic throughout (53 checks), no floats, deterministic. [E] CONSTRUCTION: |s> = (E^theta_{-1})^2|0>, theta the highest root, weight 2 theta (the top of the 27000), level 2 = 8 quarter units = q^2 (an INTEGER level); the Chevalley/Frenkel-Kac basis is MACHINE-built from the v497 root data (bimultiplicative 2-cocycle eps(a,b) = (-1)^(m_a^T B m_b), asymmetry verified on all 57600 ordered root pairs; the [e_a,e_{-a}] = SGN*H(a) sign FORCED by the Jacobi identity, SGN = -1, not chosen; the invariant form kappa DERIVED from invariance, kappa(theta_vee,theta_vee) = 2; Jacobi exact on ALL 57600 sl2-slice triples + 6000 seeded triples + all 496 theta-adjacent triples; all root-sum structure constants N_ab = +-1). [E] SINGULARITY (main theorem, machine): J^a_1|s> = 0 for ALL 248 basis elements, 0 violations; case classification of the J_1 condition: 190 vanish at the first bracket, 57 at the second, exactly 1 (a = F^theta) by the central-term cancellation -- the ONLY case that sees the level k; plus J^a_2|s> = 0 (all 248), E^a_0|s> = 0 (all 120 positive roots), h_i-eigenvalues = <2 theta, a_i_vee> exactly, Shapovalov norm 0 at k = 1 (4 at k = 2); the affine PBW engine ([J^a_m,J^b_n] = [a,b]_{m+n} + m k kappa(a,b) delta_{m+n,0}) unit-tested on all 61504 basis pairs. [E] LEVEL DIAL: the F^theta_1 coefficient is 2(1-k) -- 2 at k = 0, 0 EXACTLY at k = 1, -2 at k = 2: in the centerless loop algebra (k = 0) the deletion object does NOT exist -- the central extension is load-bearing. [E] MU4 DATA (glue frame, v492/v496): theta-grading dims (1,56,134,56,1), g_0 = e7 + u(1); the h-adapted Weyl chamber MACHINE-constructed (8 simple roots re-derived, Cartan det 1, E8 Dynkin diagram with branch lengths [1,2,4]); theta_glue the unique dominant root, height 29 = h_vee - 1, <theta,h> = 5 => glue class j(theta) = 1; clock phase on |s> = i^(2j) = -1; class(2 theta) = 2 sheet-EVEN (consistent with WP5a S6.8); the deleting theta-sl2 crosses the sheet-ODD classes (1,3) while its square is sheet-even; E^theta expressible in the sector-C_1 equivariant variables (spatial degree m = -j mod 4, 2 monomial slots) and the square is mu4-INVARIANT; quarter bookkeeping (v496 per-period dictionary): E^theta_{-1} = 4 quarter units = one full mu4 period (248 currents), |s> = 8 quarters = q^2; cross-table theta-grading x glue class consistent (rows (1,56,126,56,1), columns (52,64,60,64)). [E] LEVEL-2 MATCH: weight 2 theta has multiplicity 1 in the level-2 Fock => U(g)|s> is THE 27000; dim V(2 theta) = 27000 = 30^3 = h_vee^3 (Freudenthal + Weyl-orbit sums, v497 replication); the direct g_0-lowering orbit BFS from |s> reproduces the Freudenthal weight multiplicities of V(2 theta) EXACTLY, weight by weight, through depth 4; quotient 31124 - 27000 = 4124 = theta-split sector sum (1036,1024,1040,1024); LATTICE KERNEL READING: (2theta,2theta)/2 = 4 > 2, so the quotient (E8)_1 theory has NO momentum-2theta state at q^2 -- |s> IS the kernel generator of Fock -> E4/eta^8. [E] NEGATIVE CONTROLS (honest separation): (a) E^theta_{-1}|0> (level 1) NOT singular -- exactly one witness (a = F^theta, -> -k|0>), the 248-current state that must REMAIN (chi_1 = 248); (b) 4 seeded generic level-2 states all non-singular (14-33 witnesses each); (c) at k = 2 the square is not singular but the CUBE (E^theta_{-1})^3|0> IS -- the (E^theta)^{k+1} pattern is GENERIC Kac level mechanics, not E8-specific; (d) D8/SO(16)_1 through the SAME pipeline (Chevalley + Jacobi on all 12544 sl2-slice triples): (E^theta_{-1})^2|0> IS singular there too (the singular-vector mechanism is level-1 GENERIC), BUT dim V(2 theta_D8) = 5304 != 14^3 and D8 keeps 3 extra level-1 primaries (comark-1 weights, h = (1/2,1,1), h = 1/2 NON-integer breaking one-block fusion) vs E8 (comarks all >= 2, vacuum the ONLY level-1 primary, glue weights (0,1,1,1) integer) -- the ONE-BLOCK closure is the E8/mu4-specific part; the singular vector itself is generic level-1 Kac mechanics, honestly separated. [C] NAMED IDENTIFICATIONS: Weyl complete reducibility of the finite-dim level-2 Fock extends the depth-4 orbit agreement to the full dim U(g)|s> = 27000; 'the singular vector generates the maximal submodule' (Kac) used only beyond level 2; the per-period quarter dictionary (v496 S4) for the u-unit bookkeeping. [O] HONEST LIMIT (= the precise WP5c question): in the TWISTED quarter-SLOT moding of the chi_w limit Fock two sector-C_1 modes never sum to 8 quarters (0 solutions, minimum total 6 = q^{3/2}) -- the per-PERIOD dictionary (v496), not the per-slot identification, carries |s> to q^2; exactly the GNS/limit-state seam (kernel containing the ideal) that WP5c must construct; local normality/Haag duality (WP5d) and Costello-Li (WP5e) stay OPEN. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact construction/singularity/control arithmetic, integer/Fraction, deterministic; Chevalley basis + affine PBW engine unit-tested on all 61504 pairs) + [C] (the named identifications: Weyl complete reducibility beyond BFS depth 4; Kac mechanics beyond level 2; per-period quarter dictionary) + [O] (the GNS/limit-state question WP5c-e -- the twisted-slot moding tension is the precise WP5c question; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP5b [E]/[C]/[O]: the deletion operator EXISTS and is explicit -- |s> = (E^theta_{-1})^2|0> in a machine-built Chevalley/Frenkel-Kac basis, J^a_1|s> = 0 on all 248 generators (case tally 190/57/1, exactly one case sees k), level dial 2(1-k): only k = 1 deletes, no deletion object in the centerless loop algebra; mu4-compatible: j(theta) = 1, clock phase i^2 = -1, class(2 theta) = 2 sheet-even, 8 quarters = q^2, expressible in the sector-C_1 equivariant variables (KILL branch not triggered); U(g)|s> = V(2 theta) = 27000 = h_vee^3 (orbit BFS = Freudenthal through depth 4), quotient 4124 = sector sum; controls: the 248-current state and generic states not singular, cube at k = 2 (generic Kac mechanics), D8 contrast (5304 != 14^3, h = 1/2 breaks fusion) -- one-block closure E8/mu4-specific; open: the twisted-slot moding carries |s> only via the per-period dictionary -- the precise WP5c GNS/limit-state question; WP5c-e stay [O]; SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP4.01; SEAM.EQUIV.01; v128; v492; v496; v497
Scripts: v498_celestial_wp5b_singular_vector.py
Location: tfpt_research_contracts
Original CSV, line 563Original ledger status
CELEST.WP5C.01
WP5c OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5c, the third constructive milestone of WP5 -- 'the GNS limit state'; WP5a/v497 made the boundary limit precise, WP5b/v498 built the deleting operator, WP5c constructs the STATE whose GNS kernel contains the null ideal): exact integer/Fraction arithmetic throughout (35 checks), no floats, deterministic. [E] CONSTRUCTION: the quasi-free family omega_w -- loop sector (r = 0) = the affine k=1 vacuum n-point functions via the v498 PBW engine and the MACHINE-DETERMINED compact anti-involution theta(e_a) = -e_{-a}, theta(h) = h (the UNIQUE sign that is an anti-automorphism on ALL 61504 basis pairs; contravariance <J^x_{-1} u, v> = <u, J^{theta x}_1 v> exact); radial sector (r >= 1) = oscillator pairings x^{w r} (the exact Gibbs regulator e^{-beta E_w}, kept x-adic; rational sample x = 1/2). [E] CONVERGENCE SHARP: loop entries w-independent; radial entries x^{w r} have x-adic order w r >= w, so omega_w = omega_inf mod x^{N+1} for ALL w >= N+1 (N = 8) and NOT yet at w = N (the r = 1 modes sit at order N exactly) -- the WP5a stabilisation threshold w = n+1 holds IDENTICALLY at the state level; positivity of every omega_w checked at the one-particle level (all 36 radial modes). [E] THE KERNEL (core result, fully machine): level-2 Fock = 31124 monomials in 9361 weight blocks (dim profile 164:1, 37:240, 7:2160, 1:6960); the FULL exact Gram computed block by block: level-1 rank 248 POSITIVE DEFINITE (240 root modes at norm k, Cartan block = the E8 Cartan matrix, det 1); level-2 rank 4124 EXACTLY (the preregistered target check), kernel 27000, every one of the 9361 blocks PSD; rank per Weyl orbit (norm 8/6/4/2/0) = (0,0,1,8,44), and (dim, rank, inertia) CONSTANT along every Weyl orbit (verified on all blocks, no orbit assumption); kernel = V(2 theta) WEIGHT BY WEIGHT (rank_lambda = dim_lambda - mult_lambda(V(2 theta)) for all 9361 weights, Freudenthal cross-check); component route: unique highest-weight vectors at (0, omega, theta, 2theta) with norms (3844, 49, 2, 0); direct ideal checks F_i|s> and F_j F_i|s> norm 0 and orthogonal to their full blocks; GNS graded dims (1, 248, 4124) = E4/eta^8 through q^2. [E] MU4 EQUIVARIANCE: frame map glue -> standard via the UNIQUE E8 diagram isomorphism; omega_w o clock = omega_w for ALL w (nonzero vacuum values only at total weight 0 = clock class 0); the clock unitary DESCENDS to GNS(omega_inf); GNS level-2 rank split by clock class = (1036, 1024, 1040, 1024) = Theta_Cj/eta^8 at q^2, level-1 split (60,64,60,64) -- the WP5a two-routes identity now holds at the STATE level. RESOLUTION OF WP5B S4.10: |s> IS the ZERO VECTOR of GNS(omega_inf) -- nothing needs to 'carry' it to q^2; the twisted two-C1-mode floor (6 quarters = q^{3/2}) concerns only pre-limit objects (GNS has no state at 6 quarters: 6 mod 4 != 0). [E] NEGATIVE CONTROLS: (a) k = 2: <s|s> = +4 (WP5b sign confirmed), all 240 norm-8 and all 6720 norm-6 monomials nonzero, dominant blocks FULL rank -- the GNS kernel contains NO level-2 ideal (the deletion is a k = 1 state-level fact); (b) k = 0: level-1 Gram identically 0 -- no current layer without the central extension; (c) D8/SO(16)_1 same pipeline: level-1 rank 120 PD, FULL level-2 Gram over 7380 monomials in 2705 blocks: rank 2076 = 7380 - 5304, kernel = V(2 theta_D8) weight by weight, PSD, lattice route Theta_D8/eta^8 = (1, 120, 2076) agrees; but h = (0, 1/2, 1, 1): the vacuum GNS is 1 of 4 blocks vs E8 = the FULL character -- ONE-BLOCK completeness of the limit state is E8/mu4-specific; (d) wrong family E'_w = m w + r: level-1 limit rank 0 (the 248 layer ERASED), u^4 survivor content 710955 != 248 -- the kill has teeth; (e) no damping: the w-uniform undamped vacuum keeps the full jet Fock, 897266 != 248; (f) CCR OBSTRUCTION: on a FIXED radial pairing c > 0 every state has omega(b b*) >= c (positivity) -- NO w-uniform state damps the radial modes: the contract's family formulation is NECESSARY, and the family EXISTS => the preregistered KILL is probed and NOT triggered. [C] NAMED IDENTIFICATIONS: the quasi-free/Gaussian extension of the positive one-particle data to the full *-algebra (standard); Kac positivity beyond level 2 (levels <= 2 machine-checked here); complete reducibility / multiplicity-freeness in the component route (the rank is [E], the module identification is the [C] part); k = 2 non-dominant blocks by Weyl-orbit rank constancy. [O] HONEST LIMIT: local normality of omega_inf, the net structure / Haag duality (WP5d), Costello-Li (WP5e), and the operator-algebraic closure of the radial contraction beyond the graded/x-adic topology stay OPEN. VERDICT: SUCCESS on the preregistered criteria at the graded-state level (target check dim GNS level-2 = 4124 met exactly). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact construction/Gram/control arithmetic, integer/Fraction, deterministic; the compact anti-involution machine-determined; the full 9361-block level-2 Gram exact) + [C] (the named identifications: quasi-free extension; Kac positivity beyond level 2; module identification in the component route) + [O] (local normality, net structure/Haag duality, Costello-Li -- WP5d/e; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP5c [E]/[C]/[O]: the GNS limit state EXISTS and is explicit -- the quasi-free family omega_w (affine k=1 vacuum + x^{wr}-contracted radial oscillators; compact adjoint machine-determined on all 61504 pairs) is positive for every finite w, stabilises exactly at the WP5a threshold w = N+1, and its limit has GNS levels (1, 248, 4124): full 9361-block exact Gram, level-2 rank 4124 = chi_2, kernel 27000 = V(2 theta) weight by weight, every block PSD, rank table (0,0,1,8,44); the 248 layer survives (level-1 rank 248 PD); the clock descends to GNS with rank split (1036,1024,1040,1024) = the per-period theta-split; |s> is the GNS zero vector (WP5b S4.10 resolved at the state level); CCR obstruction: no w-uniform damping state -- the family is necessary AND exists (KILL not triggered); controls: k = 2 keeps everything, k = 0 has no current layer, D8 = one block of four, wrong family/no damping caught (710955/897266 != 248); open: local normality, net/Haag duality (WP5d), Costello-Li (WP5e); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5A.01; CELEST.WP5B.01; SEAM.EQUIV.01; v128; v492; v496; v497; v498
Scripts: v500_celestial_wp5c_gns_limit_state.py
Location: tfpt_research_contracts
Original CSV, line 564Original ledger status
CELEST.WP5D.01
WP5d-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5d, alpha stage -- 'the two-interval index from the lattice'; the local-net question): numpy/scipy Gaussian machinery + exact Fractions (39 checks). Roadmap discriminator: (E8)_1 => mu = 1 (Haag-dual two-interval net, ONE sector); SO(16)_1 => mu = 4; preregistered KILL = 'the two-interval mu-offset does NOT vanish after the condensation'. [E-machine]/[L] SETUP: the v367 seam edge as 16 chiral Majorana layers (c_- = 8 = g_car + N_fam, v489), faithful 1d reduction = the critical Majorana ring (v450 style, NS closure, c = 1/2 per layer); Peschel-exact; three two-interval prescriptions (fermionic rho / orbifold sector sum {1, P_A, P_B, P_AB} / dual algebra {1, P_AB}); the superselection identity [rho, P_AB] = 0 machine-exact; EVERY Gaussian formula (covariance round trip, Peschel entropies, trace formula, interval parity = Pfaffian, disjoint Renyi-2 deficit) validated against exact diagonalisation to <= 1.5e-15. [L/N] FERMIONIC MI EXTENSIVE (the mu = 1 reference): I(x) = -(c/3) ln(1-x) with c fitted 0.5000; max |F(x)| = 5.97e-5 at N = 512 over the full x grid, |F| falling monotonically to 1.3e-6 at N = 1024 -- the lattice witness of Casini-Huerta extensivity. [L/N] THE MU-OFFSET (core): Renyi-2 exactly Delta_2 = -ln[(1+R)/2] with R = Tr(rho P rho P)/Tr rho^2; the 16-layer seam carrier (diagonal Z2) saturates ln 2 to MACHINE PRECISION (|Delta_2 - ln 2| = 1.1e-15 at N = 512, 0 at N = 1024/2048; single-layer R falls as N^-0.500, Ising disorder decay); ED (N = 12..24) confirms vN and Renyi-3/4 deficits track it (not a Renyi-2 artefact); index reading [F : F_even] = exp(Delta_inf) = 2.000000 => mu_gauged = exp(2 Delta_inf) = 4.000000 = v490's mu_pre (TWO independent lattice witnesses: parity census there, entropic two-interval offset here); gauging every layer separately costs 16 ln 2 -- the offset measures ln[index of the subnet], not a universal blob. [L/N] HAAG-DUALITY INDICATOR: fermionic complementarity S(E) = S(E') to 8.2e-12 (graded-duality proxy); the orbifold BREAKS it (D(E) - D(E') = 0.427 at a = 64 -- the direct lattice witness that Even(E)-prime != Even(E-prime)); the complementary-pair sum is exchange-symmetric (dev 4.3e-14) and bounded by ln 4 = ln mu(SO(16)_1), max 1.2637 at the self-dual x = 1/2, -> ln 2 at the squeezed ends: the ln 4 KLM budget is a DOUBLE-LIMIT statement (sum of the two one-sided ln 2 saturations), NOT a fixed-x identity -- the first naive fixed-x check was WRONG and was replaced by the honest budget + asymmetry pair; the dual-algebra prescription {1, P_AB} is machine-identically defect-free. [E] CONDENSATION ARITHMETIC, LATTICE-ANCHORED: det Cartan(D5) * det Cartan(A3) = 4 * 4 = 16 = mu(carrier); KLM/Longo-Rehren quotients mu/|L|^2: 16/4^2 = 1 (one-step mu4 glue) and 16/2^2 = 4 -> 4/2^2 = 1 (via SO(16)_1), both routes exact Fractions; Sigma d_i^2 cross-checks (Ising 1+1+2 = 4, SO(16)_1 four abelian sectors = 4, (E8)_1 one sector = 1); theta_v = e^{2 pi i nu/16} = 1 EXACTLY at nu = 2 c_- = 16 (bosonic, condensable, v490), rivals nu = 1, 2, 8 give theta_v != 1. THE ANCHORED CHAIN: mu = 1 (graded fermion, MEASURED offset 0) -> 4 (gauged, MEASURED ln 2 offset + v490 parity census) -> 1 (theta_v = 1 extension, ARITHMETIC [C]: the condensed (E8)_1 net is interacting, not free-fermion computable -- v490's honest fence): the lattice pins BOTH measurable ends, the arithmetic closes the middle. [E/N] NEGATIVE CONTROLS: (a) nu = 1 single Majorana: offset non-trivial (Delta_2 = 0.6364 at N = 2048, rising toward ln 2) and theta_v = e^{i pi/8} != 1 -- NO bosonic condensation channel, the offset NOT removable: the KILL discriminator has teeth; (b) trivial dimerised phase: c = 0.00000, MI <= 1.3e-7, 16-layer deficit <= 2.6e-8 -- the ln 2 is a property of the CRITICAL seam edge, not of the prescription; (c) wrong sector sum ({1, P_AB} only, 2 of 4 structures): offset EXACTLY 0 instead of ln 2 -- measurably wrong bookkeeping, as preregistered. [C] the continuum dictionary 'offset = ln index' (KLM two-interval index; Longo-Xu; Casini-Huerta-Magan-Pontello entropic order parameters) and the condensation step itself (theta_v = 1 simple-current extension). [O] continuum Haag duality, local normality, split/strong additivity of the quotient net (WP5d-beta, running as exploration), Costello-Li (WP5e). VERDICT: the mu = 1 indicator chain PASSES; the preregistered KILL is NOT triggered. SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal [E] (the exact arithmetic: Cartan dets, KLM/Longo-Rehren quotients, Sigma d^2, theta_v at nu = 16, machine identities) + Lattice/Numerical [L]/[N] (the Gaussian lattice witnesses, ED-validated to 1e-15: extensivity, the ln 2 plateau, the duality-asymmetry witness) + [C] (the continuum dictionary offset = ln index -- KLM/Longo-Xu/CHMP -- and the interacting condensation step) + [O] (continuum Haag duality / local normality / split of the quotient net = WP5d-beta; Costello-Li = WP5e; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP5d-alpha [E]/[L]/[N]/[C]/[O]: the two-interval mu-index measured entropically on the seam lattice edge -- fermionic MI extensive (mu = 1 reference, c fit 0.5000, |F| -> 0), the sector-summed orbifold pays exactly one classical bit (16-layer carrier ln 2 plateau to 1.1e-15 at N = 512), [F : F_even] = 2, mu_gauged = 4 = the v490 census (two independent lattice witnesses); complementary-pair budget <= ln 4 (double-limit statement) + the orbifold duality-failure witness; condensation arithmetic anchored at both measured ends: 16 -> 4 -> 1 (KLM/Longo-Rehren, Sigma d^2, theta_v = 1 exactly at nu = 16, rivals != 1) => mu = 1 after condensation, preregistered KILL not triggered; controls: nu = 1 non-removable, trivial phase zero, wrong sector sum loses the full ln 2; open: WP5d-beta (continuum Haag duality / local normality / split, exploration running) + WP5e (Costello-Li); SEAM.EQUIV.01 stays [O]
Dependencies: CELEST.SEAM.01; CELEST.WP5C.01; SEAM.CCC.01; SEAM.MU.01; SEAM.EQUIV.01; v367; v378; v450; v489; v490
Scripts: v501_celestial_wp5d_two_interval_index.py
Location: tfpt_research_contracts
Original CSV, line 565Original ledger status
CELEST.WP5E.ALPHA.01
WP5e-ALPHA OF THE CELESTIAL-HOLOGRAPHIC CONTRACT EXECUTED (2026-07-21; CELEST.SEAM.01 work package 5e, alpha stage -- 'prefactor and level bookkeeping', the CFT-side anchor of the BCOV/Costello-Li twistor uplift): exact sympy/Fraction arithmetic, no floats (33 checks). Roadmap target: boundary partition function = E4/eta^8 INCLUDING the q^{-1/3} prefactor; preregistered KILL = 'the anomaly inflow demands a level != 1'. [E] PART A -- THE PREFACTOR IS EXACT mu4 VACUUM-ENERGY BOOKKEEPING: Hurwitz-zeta derivation (zeta(-1,theta) = -B2(theta)/2 symbolic; E_b(theta) = -1/24 + theta(1-theta)/4 exact; fermion counterpart E_f = -E_b, Ising h_sigma = 1/16 per Majorana); the clock is INNER -- h = (2,2,2,2,2;0^4) and the A3 detector h' = (0^5;1,1,1,-3) BOTH read the glue class mod 4 on all 240 roots ((h,h) = 20, (h',h') = 12, sum 32 = 2*4^2) -- so ALL 8 Cartan/torus-boson directions sit at clock eigenvalue 1: twist distribution theta = 0^8 in EVERY sector C_j, a SHIFT orbifold (lattice cosets, integer-moded oscillators), NOT a rotation orbifold; E_osc(C_j) = 8*(-1/24) = -1/3 = -c/24 SECTOR-INDEPENDENT (c = 8 = rank = Sugawara 248/31 -- the central-charge route and the boson-count route agree); sector ground exponents -1/3 + (0,1,1,1). [E] DISCRIMINANT FORM = CASIMIR ENERGY, TWO ROUTES: coset minima h_D5 = (0,5/8,1/2,5/8), h_A3 = (0,3/8,1/2,3/8), glue diagonal (0,1,1,1) INTEGER; route 1 = SPECTRAL FLOW (s_j = j h/4 costs j^2(h,h)/32: 5j^2/8, 3j^2/8, sum j^2 = 0 mod 1 = the isotropy/Lagrangian condition); route 2 = FREE FERMIONS, exact not just mod 1 (D5 = 10 Majorana -> 5/8, A3 = D3 = 6 -> 3/8, glue diagonal = 16 = 2^(g_car-1) = the WP5d seam carrier -> 1; R-NS shift 1/16 per Majorana); HONEST SHARPENING (i): the ROTATION reading fails -- the geometric A3 deck diag(i,i^-1) gives twisted Casimir 3/16 != 3/8 (the v492 'clock != deck' lesson at the Casimir level, factor 2). [E] CHARACTER RECONSTRUCTION: all glue-coset norms even (0 odd cross-terms: isotropic diagonal <=> even extended lattice = E8); sum of the four Theta_Cj = E4 = (1,240,2160,6720); sector characters have leading exponents -1/3 + h_j, leading coefficients (1,64,60,64), currents (60,64,60,64), and sum EXACTLY to E4/eta^8 = q^{-1/3}(1,248,4124,34752). [E] PART B -- k = 1 FORCED THREE INDEPENDENT WAYS: (i) the CURRENT CONDITION -- simple-current weights are exact closed forms (D5: h(k om_v) = k/2, h(k om_{s,c}) = 5k/8; A3: h(k om_j) = k j(4-j)/8), the Z4 extension closes onto the E8 ADJOINT (60+64+60+64 = 248 spin-1 currents) only if h(J) = 1, and h(J; k) = k => k = 1 uniquely; HONEST SHARPENING (ii): glue-h INTEGRALITY h(J^j; k) = k(0,1,1,1) holds for ALL k = 1..8 -- the originally conjectured integrality route fixes NOTHING (the naive dial is dead; the three other dials carry); (ii) CONFORMAL EMBEDDING: D5+A3 in E8 reduces to 47k^2 + 219k - 266 = 47(k-1)(k+266/47) = 0 (unique positive root k = 1), D8 in E8 to 128k(1-k) = 0 (128 = spinor dim); (iii) CENTRAL CHARGE = THE PREFACTOR ITSELF: 248k/(k+30) = 8 <=> 240(k-1) = 0 (240 = #roots), -c/24 = -1/3 only at k = 1; PLUS the WP5b singular-vector dial: PBW engine ||(E^theta_{-1})^m 0||^2 = m! k(k-1)...(k-m+1), singular vector EXACTLY at grade k+1, Weyl dims from root data (248, 3875, 27000; Sym^2 248 = 30876), k = 1 deletes V(2 theta): 31124 - 27000 = 4124 = chi_2 (singular-vector consistency); k = 2: <s|s> = +4 = the WP5b/WP5c value. [E] NEGATIVE CONTROLS: (a) D8/SO(16)_1 -- same rank 8, same c = 8, SAME q^{-1/3} prefactor, but coset weights (0,1/2,1,1): the prefactor does NOT discriminate E8 vs SO(16), the h-INTEGRALITY of the glue diagonal does; Theta_D8 + Theta_s = E4 (the {1,s} Lagrangian extension IS E8); (b) WRONG TWIST: Z2 rotation gives E = +1/6 (h = 1/2 -- no mu4-shift sector there); a mu4 rotation makes E_j = (-1/3, 1/24, 1/6, 1/24) sector-DEPENDENT (the common prefactor and the single eta^8 break); (c) WRONG k in {2,3,4}: ALL FIVE dials fail coherently (h(J) = k != 1; embedding residuals 360/814/1362; c != 8; prefactor != q^{-1/3}; <s|s> = 4/12/24 so 27000 survives, level-2 dim stays 31124 != 4124); at k = 1 all five pass. [C] Kac generation of the maximal submodule by the grade-(k+1) singular vector for k >= 3 (k = 1, 2 machine-anchored by WP5b/WP5c). [O] HONEST FENCE -- WP5e is NOT closed: the BCOV/Costello-Li construction itself (the boundary partition function from the twistor side, the actual anomaly-inflow computation on PT, Costello's one-loop axion coupling lambda_g/(8 pi sqrt 3), the Costello-Paquette-Sharma level-from-flux mechanism) stays OPEN; v495 anchors the Lie-algebra arithmetic of the axion coefficient only; the KILL branch is NOT triggered on the CFT side -- all computable dials say k = 1 -- but its twistor side is untested. v502 anchors the CFT side of WP5e (alpha stage). SEAM.EQUIV.01 stays [O], untouched. No gate closes, no marker moves.
Recorded status: Formal/Identity [E] (exact prefactor/level arithmetic: Hurwitz-zeta vacuum energies, inner/shift twist distribution, discriminant-form = Casimir two-route identity, current/embedding/central-charge k-fixing, singular-vector dial -- sympy/Fraction exact, Wolfram-mirrored) + [C] (Kac generation for k >= 3) + [O] (the BCOV/Costello-Li twistor inflow = WP5e proper; SEAM.EQUIV.01 unchanged)
Dependencies, status detail and source
Canonical status: WP5e-alpha [E]/[C]/[O]: the q^{-1/3} prefactor of E4/eta^8 derived as exact mu4 vacuum-energy bookkeeping (inner clock => shift orbifold theta = 0^8, common -c/24 = -1/3 at c = 8; discriminant form = Casimir via spectral flow AND 16-Majorana free fermions; rotation reading fails 3/16 != 3/8) and k = 1 forced three independent ways (current condition h(J) = k = 1; conformal embedding 47(k-1)(k+266/47) = 0 resp. 128(k-1) = 0; c = 8 <=> 240(k-1) = 0; plus the WP5b singular-vector dial 31124 - 27000 = 4124) with the honest sharpening that glue-h integrality holds for ALL k and fixes nothing; controls: D8 same prefactor (discriminator = h-integrality), rotation twists break the common prefactor, wrong k fails all five dials; the BCOV/twistor inflow itself stays [O] -- WP5e not closed, SEAM.EQUIV.01 untouched
Dependencies: CELEST.SEAM.01; CELEST.WP1.01; CELEST.WP4.01; CELEST.WP5A.01; CELEST.WP5B.01; CELEST.WP5C.01; CELEST.WP5D.01; SEAM.EQUIV.01; v492; v496; v497; v498; v500; v501
Scripts: v502_celestial_wp5e_prefactor_level.py
Location: tfpt_research_contracts
Original CSV, line 566